Group von Neumann Algebras and Related Algebras

Dissertation

zur Erlangung des Doktorgrades

der Mathematisch-Naturwissenschaftlichen Fakultäten der Georg-August-Universität zu Göttingen

vorgelegt von

Holger Reich

aus Veerßen

Göttingen 1998

D7 Referent: Prof. Dr. W. Lück Korreferent: Prof. Dr. T. tom Dieck Tag der mündlichen Prüfung: 28.1.1999

Contents

1 Introduction 1 1.1 Notations and Conventions . . . . . . . . . . . . . . . . . . . . 5

2 The Algebra of Operators Affiliated to a Finite von Neumann Algebra 6

3 Dimensions 12 3.1 A-modules, U-modules and Hilbert A-modules . . . . . . . . . 12 3.2 A Notion of Dimension for U-modules . . . . . . . . . . . . . . 18 3.3 The Passage from A-modules to U-modules . . . . . . . . . . 23

4 New Interpretation of L^{2}-Invariants 29

5 The Atiyah Conjecture 31 5.1 The Atiyah Conjecture . . . . . . . . . . . . . . . . . . . . . . 31 5.2 A Strategy for the Proof . . . . . . . . . . . . . . . . . . . . . 35 5.3 The Relation to the Isomorphism Conjecture in Algebraic K-

theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6 Atiyah’s Conjecture for the Free Group on Two Generators 45 6.1 Fredholm Modules . . . . . . . . . . . . . . . . . . . . . . . . 45 6.2 Some Geometric Properties of the Free Group . . . . . . . . . 48 6.3 Construction of a Fredholm Module . . . . . . . . . . . . . . . 49

7 Classes of Groups and Induction Principles 52 7.1 Elementary Amenable Groups . . . . . . . . . . . . . . . . . . 52 7.2 Linnell’s Class C . . . . . . . . . . . . . . . . . . . . . . . . . 55

8 Linnell’s Theorem 62 8.1 Induction Step: Extensions - The (A)-Part . . . . . . . . . . . 66 8.2 Induction Step: Extensions - The (B)-Part . . . . . . . . . . . 83 8.3 Induction Step: Directed Unions . . . . . . . . . . . . . . . . . 86 8.4 Starting the Induction - Free Groups . . . . . . . . . . . . . . 90 8.5 Starting the Induction - Finite Extensions of Free Groups … 92

9 Homological Properties and Applications 94 10 K-theory 98

10.1 K_{0} and K_{1} of A and U . . . . . . . . . . . . . . . . . . . . . . 98 10.2 Localization Sequences . . . . . . . . . . . . . . . . . . . . . . 103

11 Appendix I: Affiliated Operators 106

12 Appendix II: von Neumann Regular Rings 116

13 Appendix III: Localization of Non-commutative Rings 119 13.1 Ore Localization . . . . . . . . . . . . . . . . . . . . . . . . . 119 13.2 Universal Localization . . . . . . . . . . . . . . . . . . . . . . 122 13.3 Division Closure and Rational Closure . . . . . . . . . . . . . 124 13.4 Universal R-Fields . . . . . . . . . . . . . . . . . . . . . . . . 127 13.5 Recognizing Universal Fields of Fractions . . . . . . . . . . . . 130 13.6 Localization with Respect to Projective Rank Functions …. 131

14 Appendix IV: Further Concepts from the Theory of Rings 135 14.1 Crossed Products . . . . . . . . . . . . . . . . . . . . . . . . . 135 14.2 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

1 Introduction

In their first Rings of Operators paper from 1936 Murray and von Neumann not only introduced those algebras which are nowadays known as von Neumann algebras but they also considered for the first time a notion of dimension which is not necessarily integer valued [63]. Since then this idea has found a lot of applications. In particular in 1976, studying indices of elliptic operators on coverings, Atiyah was led to define the so called L^{2}-Betti numbers which are a priori real valued [1]. It turned out later that these numbers are in fact topological and even homotopy invariants [24]. Meanwhile they admit of a definition completely analogous to the usual Betti numbers [52]. The only difference is that whereas usual Betti numbers are ranks of ℤmodules the L^{2}-Betti numbers are dimensions of modules over the group von Neumann algebra N Γ. Here Γ can for example be the fundamental group. In order to compute the usual Betti numbers it is often easier to pass to rational homology and to compute vector space dimensions. In this thesis we investigate a similar passage for von Neumann algebras. The algebra of operators UΓ affiliated to the von Neumann algebra N Γ, which was already introduced by Murray and von Neumann [63], plays the role of a quotient field for N Γ. We will show that there is a notion of dimension for arbitrary UΓ-modules which is compatible with the N Γ-dimension developed by Lück in [52]. Moreover, the algebras UΓ are large enough to host similar quotient fields DΓ for the group algebras ℂΓ. Linnell made use of this idea to prove refined versions of the zero divisor conjecture for a large class of groups [48]. It also leads to a natural explanation of the fact that in all known cases the L^{2}- Betti numbers are rational numbers. Investigating these intermediate rings we show a kind of universal coefficient theorem for L^{2}-homology and obtain information about Euler characteristics of groups. We also try to clarify the relationship to the Isomorphism Conjecture in algebraic K-theory [27]. We will now explain the results in greater detail. Let Γ be a group and let ℂΓ be the complex group ring. We investigate the following commutative square of rings, where all maps are unit-preserving embeddings of subrings.

[figure: ℂΓ; N Γ; DΓ; UΓ]

One should think of the upper horizontal map as a completion and of the vertical maps as localizations. We start at the upper right corner: Let l^{2}Γ be a complex Hilbert space with orthonormal basis the set Γ. This Hilbert space carries a natural Γ-action and this yields an embedding of the complex group ring ℂΓ into the algebra B(l^{2}Γ) of bounded linear operators.

Definition 1.1. The group von Neumann algebra N Γ is the closure of ℂΓ in B(l^{2}Γ) with respect to the weak operator topology.

We consider N Γ as a ring extension of ℂΓ. In particular we can study ℂΓ-modules by applying the functor − ⊗_{ℂΓ} N Γ in order to treat them in the category of N Γ-modules. This may seem unwise at first glance, but we even go a step further and consider a larger ring containing also unbounded operators.

Definition 1.2. The algebra UΓ of operators affiliated to the group von Neumann algebra N Γ consists of all closed, densely defined (unbounded) operators which commute with the right action of Γ on l^{2}Γ.

For details see Section 2 and Appendix I. The algebra UΓ is an Ore localization of N Γ, see Proposition 2.8. From the ring theoretical point of view it is a beautiful ring, namely it is a von Neumann regular ring (see 2.4), i.e. its Tor-dimension vanishes. In particular the category of finitely generated projective UΓ-modules is abelian. In Theorem 3.12, we show the following.

Theorem 1.3. There is a well-behaved notion of dimension for arbitrary UΓ-modules which is compatible with the N Γ-dimension studied in [52].

The advantage of working with modules in the algebraic sense in contrast to Hilbert spaces is that all tools and constructions from (homological) algebra, like for example spectral sequences or arbitrary limits and colimits, are now available. The L^{2}-Betti numbers can be read from the homology with twisted coefficients in UΓ, see Proposition 4.2. More generally we will show that the natural map

K_{0}(N Γ) → K_{0}(UΓ)

is an isomorphism, see Theorem 3.8. Finer invariants like the Novikov-Shubin invariants (see [67], [25], [54]) do not survive the passage from N Γ to UΓ. Twisted homology with coefficients in UΓ is therefore the algebraic analogue of the reduced L^{2}-homology in the Hilbert space set-up, where the homology groups are obtained by taking the kernel modulo the closure of the image of a suitable chain complex of Hilbert spaces. One of the main open conjectures about L^{2}-homology goes back to a question of Atiyah in [1].

Conjecture 1.4 (Atiyah Conjecture). The L^{2}-Betti numbers of the universal covering of a compact manifold are all rational numbers.

More precisely, these numbers should be related to the orders of finite subgroups of the fundamental group, see 5.1. We will give an algebraic reformulation of this conjecture in Section 5 and work out how it is related to the isomorphism conjecture in algebraic K-theory [27]. In [48] Linnell proves this conjecture for a certain class C of groups which contains free groups and is closed under extensions by elementary amenable groups. He examines an intermediate ring DΓ of the ring extension ℂΓ ⊂ UΓ.

Definition 1.5. Let DΓ be the division closure of ℂΓ in UΓ, i.e. the smallest division closed intermediate ring, compare Definition 13.14.

Linnell shows that for groups in the class C which have a bound on the orders of finite subgroups these rings are semisimple. We will examine his proof in detail in Section 6 and Section 8. We will emphasize that for groups in the class C the ring DΓ is a localization of ℂΓ, see Theorem 8.3 and Theorem 8.4. But for non-amenable groups one has to replace the Ore localization by a universal localization in the sense of Cohn, compare Appendix III. Contrary to Ore localization universal localization need not be an exact functor. In Theorem 9.1, we will show the following result.

Theorem 1.6. For groups in the class C with a bound on the orders of finite subgroups we have

Tor_{p}^{ℂΓ}(−; DΓ) = 0 for p ≥ 2.

This leads to a universal coefficient theorem for L^{2}-homology (Corollary 9.2) and to the following result, compare Corollary 9.3.

Corollary 1.7. The group Euler characteristic for groups in the class C with a bound on the orders of finite subgroups is nonnegative.

In Section 7 we will investigate the class C and show that it is closed under certain processes, e.g. under taking free products. For instance P SL_{2}(ℤ) belongs to the class C. In Section 10 we will collect results about the algebraic K-theory of our rings ℂΓ, DΓ, N Γ and UΓ. In particular we compute K_{0} of the category of finitely presented N Γ-modules, see Proposition 10.10. Several Appendices which collect material mostly from ring theory will hopefully make the results more accessible to readers with a different background. To give the reader an idea we will now examine the easiest non-trivial examples.

Example 1.8. Let Γ = ℤ be an infinite cyclic group. We can identify the Hilbert space l^{2}Γ via Fourier transformation with the space Ł^{2}(S^{1}, μ) of square integrable functions on the unit circle S^{1} with respect to the usual measure. If we think of S^{1} as embedded in the complex plane we can identify the group algebra ℂℤ with the algebra of Laurent polynomials ℂ [z^{±1}] considered as functions on S^{1}. A polynomial operates on Ł^{2}(S^{1}, μ) by multiplication. The von Neumann algebra can be identified with the algebra Ł^{∞}(S^{1}, μ) of (classes of) essentially bounded functions. The algebra of affiliated operators is in this case the algebra Ł(S^{1}) of (classes of) all measurable functions. The division closure of ℂ [z^{±1}] in Ł(S^{1}) is the field ℂ(z) of rational functions. The diagram on page 2 becomes in this case

[figure: ℂ [z^{±1}]; Ł^{∞}(S^{1}, μ); ℂ(z); Ł(S^{1}).]

Note that the inverse of a Laurent polynomial which has a zero on S^{1} is not bounded, but of course it is a measurable function. We obtain ℂ(z) from ℂ [z^{±1}] by inverting all non-zerodivisors, i.e. all nontrivial elements.

Example 1.9. Let Γ = ℤ ∗ ℤ be the free group on two generators. This example is already much more involved. It will turn out that DΓ again is a skew field. The proof of this fact uses Fredholm module techniques and will be presented in Section 6, compare also Theorem 5.16. This time DΓ is a universal localization (and even a universal field of fractions) of ℂΓ in the sense of Cohn [15], compare Appendix III. The functor − ⊗_{ℂΓ} DΓ is no longer exact.

1.1 Notations and Conventions

All our rings are associative and have a unit. Ring homomorphisms are always unit-preserving. For a ring R we denote by R^{×} the group of invertible elements in the ring. M(R) denotes the set of matrices of arbitrary (but finite) size. GL(R) is the set of invertible matrices. We usually work with right-modules since these have the advantage that matrices which represent right linear maps between finitely generated free modules are multiplied in the way the author learned as a student. An idempotent in a ring R is an element e with e^{2} = e. We talk of projections only if the ring is a ∗-ring. They are by definition elements p in the ring with p = p^{2} = p^{∗}. A non-zerodivisor in a ring R is an element which is neither a left- nor a right-zerodivisor. A union ⋃_{i∈I} M_{i} is called directed if for i and j in I there always exists a k ∈ I such that M_{i} ∪ M_{j} ⊂ M_{k}. More notations can be found in the glossary on page 140.

2 The Algebra of Operators Affiliated to a Finite von Neumann Algebra

We will now introduce one of our main objects of study: The algebra UΓ of operators affiliated to the group von Neumann algebra N Γ. We will show that UΓ is a von Neumann regular ring and that it is a localization of N Γ. Even though we are mainly interested in these algebras associated to groups it is convenient to develop a large part of the following more generally for finite von Neumann algebras. Let B(H) be the algebra of bounded operators on the Hilbert space H. A von Neumann algebra is a ∗-closed subalgebra of B(H) which is closed with respect to the weak (or equivalently the strong) operator topology. The famous double commutant theorem of von Neumann says that one could also define a von Neumann algebra as a ∗-closed subalgebra with the property A′′ = A. Here M′ for a subset M ⊂ B(H) is the commutant

M′ = {a ∈ B(H) | am = ma for all m ∈ M}.

Recall that a von Neumann algebra A is finite if and only if there is a normal faithful trace, i.e. a linear function tr_{A} : A → ℂ, so that tr_{A}(ab−ba) = 0 and tr_{A}(a^{∗}a) = 0 implies a = 0. Normality means that the trace is continuous with respect to the ultraweak topology. Equivalently given an increasing net p_{λ} of projections lim_{λ} tr_{A}(p_{λ}) = tr_{A}(lim_{λ} p_{λ}). This is the non-commutative analogue of the monotone convergence theorem in integration theory. Note that an increasing net of projections p_{λ} converges strongly to the projection onto the closure of the subspace generated by the p_{λ}(H), and since a von Neumann algebra is strongly closed this projection lies in A. This continuity property of the trace will be crucial later, when we develop a dimension function for arbitrary U-modules. Note that the trace is not unique. However if A = N Γ is a group von Neumann algebra we will always use the trace given by a ↦< ae, e > where e ∈ Γ ⊂ l^{2}Γ is the unit of Γ.

Definition 2.1. Given a finite von Neumann algebra A we denote by U be the set of all (unbounded) operators a = (a, dom(a)) which satisfy the following three conditions:

(i) a is densely defined, i.e. dom(a) is dense in H.

(ii) a is closed, i.e. its graph is closed in H ⊕ H.

(iii) a is affiliated, i.e. for every operator b ∈ A′ we have

ba ⊂ ab.

If A = N Γ is a group von Neumann algebra we write UΓ instead of U.

Here c ⊂ d means that the domain of the operator c is contained in the domain of d, and restricted to this smaller domain the two coincide. For more information on bounded and unbounded operators we refer to Appendix I. Note that an operator a ∈ U which is bounded lies in A by the double commutant theorem. Therefore A is a subset of U. We define the sum and product of two operators a, b ∈ U as the closure of the usual sum and product of unbounded operators. It is not obvious that these closures exist and lie in U, but in fact even much more holds.

Theorem 2.2. The set U together with these structures is a ∗-algebra which contains the von Neumann algebra A as a ∗-subalgebra.

Proof. This is already proven in the first Rings of Operators paper by Murray and von Neumann [63]. We reproduce a proof in Appendix I.

Example 2.3. Let X ⊂ ℝ be a closed subset. Let A = Ł^{∞}(X, μ) be the von Neumann algebra of essentially bounded functions on X with respect to the Lebesgue measure μ. This algebra acts on the Hilbert space Ł^{2}(X, μ) of square integrable functions by multiplication. The associated algebra U of affiliated operators can be identified with the algebra Ł(X; μ) of all (classes of) measurable functions on X [73, Chapter 5, Proposition 5.3.2].

Note that on U there is no reasonable topology anymore. So U does not fit into the usual framework of operator algebras. The following tells us that we have gained good ring theoretical properties.

Proposition 2.4. The algebra U is a von Neumann regular ring, i.e. for every a ∈ U there exists b ∈ U so that aba = a.

Proof. Every a ∈ U has a unique polar decomposition a = us where u is a partial isometry in A and s ∈ U is nonnegative and selfadjoint. We have u^{∗}us = s, compare Proposition 11.5 and Proposition 11.9. So if we can solve the equation sts = s with some t ∈ U we are done because with b = tu^{∗} we get

aba = ustu^{∗}us = usts = us = a.

Define a function f on the real half line by

f(λ) = \{ \begin{matrix}0 if λ = 0 \\ \frac{1}{λ} if λ > 0 \end{matrix} .

The function calculus 11.4 tells us that sf(s)s = s.

This simple fact has a number of consequences. In Appendix II we collect several alternative definitions of von Neumann regularity and a few of the most important properties. From homological algebra’s point of view von Neumann regular rings constitute a distinguished class of rings. They are exactly the rings with vanishing weak- (or Tor-) dimension.

Note 2.5. A ring R is von Neumann regular if and only if every module is flat, i.e. for every R-module M the functor − ⊗_{R} M is exact.

Proof. Compare 12.1.

Passing from A to U more operators become invertible since unbounded inverses are allowed.

Lemma 2.6. Let a ∈ U be an operator. The following statements are equivalent.

(i) a is invertible in U.

(ii) a is injective as an operator, i.e. ker(a : dom(a) → H) = 0.

(iii) a has dense image, i.e. \overline{im(a)} = \overline{a(dom(a))} = H.

(iv) l_{a} : U → U given by b ↦ ab is an isomorphism of right U-modules.

If moreover a ∈ A ⊂ U the above statements are also equivalent to:

(v) a is a non-zerodivisor in A.

(vi) l_{a} : A → A, b ↦ ab is injective.

Proof. (i) ⇔ (ii) ⇔ (iii): We only show (ii) ⇒ (i) and (iii) ⇒ (i). Let a = us be the polar decomposition of a. Here u is a partial isometry and hence p = u^{∗}u and q = uu^{∗} are projections. We have

\overline{im(a)} = im(u) = im(uu^{∗}) = im(q),  and
ker(a) = ker(u) = ker(u^{∗}u) = ker(p).

Now if \overline{im(a)} = H, then q = id and 0 = tr_{A}(id) − tr_{A}(q) = tr_{A}(id) − tr_{A}(p) = tr_{A}(id − p). Since the trace is faithful this implies p = id and therefore u is a unitary operator and in particular invertible. In the case ker(a) = 0 we argue similarly. Now since ker(s) = ker(u) = 0 we can use the function calculus (Proposition 11.4) to define an inverse f(s)u^{∗} with f(λ) = \frac{1}{λ} for λ ≠ 0.

(i) ⇔ (iv) is clear.

(i) ⇒ (v) and (v) ⇒ (vi) are easy. It remains to show (vi) ⇒ (ii). Suppose the bounded operator a ∈ A has a nontrivial kernel. If p_{ker(a)} denotes the projection onto the kernel we have ap_{ker(a)} = 0 and we see that l_{a} is not injective.

An operator with these properties is also called a weak isomorphism. The function calculus also allows us to write every operator as a fraction.

Lemma 2.7. Every b ∈ U can be written in the form

b = at^{−1}

with a, t ∈ A and t invertible in U.

Proof. Let b = us be the polar decomposition of b (compare Proposition 11.5). We want to replace s by an invertible operator and use the function calculus. Let p be the projection 1 − u^{∗}u. Since uu^{∗}u = u we have b = us = u(p + s). Since ker(s) = ker(us) = ker(u^{∗}u) the projection onto the kernel of s is p. In particular s′ = p + s is a selfadjoint positive operator with trivial kernel. Now take for instance

f(λ) = \{ \begin{matrix}λ & λ ≤ 1 \\ 1 & λ > 1 \end{matrix} and g(λ) = \{ \begin{matrix}1 & λ ≤ 1 \\ \frac{1}{λ} & λ > 1 \end{matrix} .

Then f(s′) and g(s′) are bounded operators affiliated to A and

us = us′ = uf(s′)g(s′)^{−1}

with uf(s′), g(s′) ∈ A and g(s) invertible in U.

Now we show that U is an Ore localization of A. For the relevant definitions we refer to the first subsection of Appendix III.

Proposition 2.8. An operator in A is a non-zerodivisor if and only if it is invertible in U. The ring A satisfies both Ore conditions with respect to the set T of all non-zerodivisors and the (classical) ring of fractions AT^{−1} is isomorphic to the algebra of affiliated operators U.

Proof. The first statement was already shown in Lemma 2.6. Now suppose (a, t) ∈ A × T is given. By the preceding Lemma we can write t^{−1}a ∈ U as a′t′^{−1} with (a′, t′) ∈ A × T. This implies the right Ore condition for the pair (A, T). Applying the anti-isomorphism ∗ : U → U yields the right handed version. Now since the inclusion A ⊂ U is T -inverting, the universal property of AT^{−1} gives us a map AT^{−1} → U. Using again Lemma 2.7 one verifies that this map is an isomorphism, compare Proposition 13.17(ii).

Corollary 2.9. The ring U is flat over A, i.e. the functor − ⊗_{A} U is exact.

Proof. Ore localization is an exact functor. Compare 13.6.

In the next section we will need the following strengthening of von Neumann regularity.

Proposition 2.10. The algebra U of operators affiliated to a finite von Neumann algebra is a unit regular ring, i.e. for every element a ∈ U there exists a unit b ∈ U^{×} with aba = a.

Proof. We have to modify the proof given for von Neumann regularity above (2.4). Let a = us be the polar decomposition. Let p be the projection uu^{∗} and q = u^{∗}u. By definition, this means that p and q are Murray von Neumann equivalent (p ∼_{MvN} q). Note that p, q, u and u^{∗} are bounded and therefore in A. In a finite von Neumann algebra p ∼_{MvN} q implies 1−p ∼_{MvN} 1−q. Compare [82, Chapter V, Proposition 1.38]. (This follows also because one knows that p ∼_{MvN} q holds if and only if tr_{Z(A)}(p) = tr_{Z(A)}(q), where tr_{Z(A)} denotes the center valued trace [41, Proposition 8.4.8 on page 532].) Therefore there exists a partial isometry v with 1 − p = vv^{∗} and 1 − q = v^{∗}v. Now u^{∗} + v^{∗} is an isometry and in particular invertible since

(u^{∗} + v^{∗})(u + v) = u^{∗}u + v^{∗}u + u^{∗}v + v^{∗}v = q + 1 − q = 1

and similar for (u + v)(u^{∗} + v^{∗}). Here we used the orthogonal decomposition H = pH ⊕ (1 − p)H = im u ⊕ ker v^{∗} = ker u^{∗} ⊕ im v to check that v^{∗}u = u^{∗}v = 0. Moreover we have (u^{∗} + v^{∗})u = u^{∗}u and we get with b = t(u^{∗} + v^{∗})

aba = ust(u^{∗} + v^{∗})us = usts.

It remains to be found an invertible t ∈ U with sts = s. So take

f(λ) = \{ \begin{matrix}1 if λ = 0 \\ \frac{1}{λ} if λ > 0 \end{matrix} .

The function calculus implies sf(s)s = s.

Another useful refinement of von Neumann regularity which is very natural in the context of operator algebras is ∗-regularity.

Note 2.11. The ∗-algebra U is a ∗-regular ring, i.e. it is a von Neumann regular ∗-ring in which a^{∗}a = 0 implies a = 0.

Proof. If a is densely defined and closed the domain of a^{∗}a is a common core for a and a^{∗}a, i.e. a can be reconstructed (as the closure) from its restriction to dom(a^{∗}a). Therefore 0 =< a^{∗}a(x), x >=< a(x), a(x) >= |a(x)|^{2} for all x ∈ dom(a^{∗}a) implies a = 0. Compare [73, Theorem 5.1.9].

For more on unit regular and ∗-regular rings see Appendix II.

3 Dimensions

The main aim of this section is to prove that there is a well-behaved notion of dimension for arbitrary U-modules (Theorem 3.12), and that −⊗_{A} U induces an isomorphism in K_{0} (Theorem 3.8). As in the preceding section A denotes a finite von Neumann algebra and U the algebra of operators affiliated to A.

3.1 A-modules, U-modules and Hilbert A-modules

On our way to a dimension for arbitrary modules we start with finitely generated ideals of the ring U itself, pass to finitely generated projective modules, then to finitely generated modules and finally to arbitrary modules. In this subsection we mainly deal with the first step. But it is also preparatory for the passage to finitely generated modules. The results for finitely generated projective modules can also be formulated in terms of A-modules or of so called Hilbert A-modules. To clarify the relation between these different approaches we systematically deal with all of them. It is convenient to organize all submodules of a given finitely generated projective module in one object: The lattice of submodules. A lattice is a partially ordered set (L, ≤) in which any two elements {x, y} have a least upper bound and a greatest lower bound. A lattice is called complete if for every subset S ⊂ L there exists the least upper bound denoted by sup(S) ∈ L and the greatest lower bound inf(S) ∈ L. Let L and L′ be complete lattices. Suppose f : L → L′ is an order isomorphism of partially ordered sets, then it is also a lattice isomorphism in the sense that f(sup(S)) = sup(f(S)) and f(inf(S)) = inf(f(S)). Before we give some examples let us recall the definition of a Hilbert Amodule: The left regular representation for group von Neumann algebras has an analogue for an arbitrary finite von Neumann algebra. Let l^{2}(A) = l^{2}(A; tr) be the Hilbert space completion of A with respect to the inner product given by

< a, b >= tr(b^{∗}a).

(This is known as the GNS construction.). There are natural left and right A-module structures on l^{2}(A) which give rise to embeddings

A → B(l^{2}(A)) and A^{op} → B(l^{2}(A)).

It turns out that with respect to these embeddings A′ = A^{op} and (A^{op})′ = A. We will always consider A as a subalgebra of B(l^{2}(A)). So for us l^{2}(A) is the defining Hilbert space for A.

Definition 3.1. A finitely generated Hilbert A-module is a Hilbert space H together with a continuous right A-module structure such that there exists a right A-linear isometric embedding of H onto a closed subspace of l^{2}(A)^{n} for some n ∈ ℕ. A morphism of Hilbert A-modules is a bounded right A-linear map.

Here is now the list of examples of lattices:

(i) The lattice L_{P roj}(A) of projections in a von Neumann algebra A with order given by p ≤ q iff qp = p. This partial order coincides with the usual order on positive operators ([62, Theorem 2.3.2]). It is well known that this lattice is complete ([82, Chapter V, Poposition 1.1])

(ii) For an arbitrary ∗-ring R the set of projections is only a partially ordered set.

(iii) Let M be a finitely generated Hilbert-A-module. Then the set of closed Hilbert A-submodules ordered by inclusion forms a lattice L_{Hilb}(M). If M = l^{2}(A) is the defining Hilbert space for the von Neumann algebra A, then this lattice is known as the lattice of Hilbert subspaces affiliated to A, i.e K ⊂ l^{2}(A) is affiliated if the corresponding projection p_{K} belongs to A (operating from the left).

(iv) Given an arbitrary ring R and a right module M_{R} one can consider the lattice of all submodules L_{all}(M_{R}). This lattice is complete. Supremum and infimum correspond to sum respectively intersection of modules.

(v) Let R be a coherent ring (compare 14.5) and let P_{R} a finitely generated projective right R-module. Then the set L_{fg}(P_{R}) of all finitely generated submodules forms a sublattice of the lattice L_{all}(P_{R}). So sup{M, N} and inf{M, N} again correspond to sum and intersection of modules. The important point here is, that over a coherent ring the intersection of two such finitely generated submodules is again finitely generated, compare [81, Chapter I, Proposition 13.3]. In particular, all this applies to a von Neumann regular ring. In that case all finitely generated submodules are direct summands. If R is a semihereditary ring (e.g. A) all finitely generated submodules are projective, compare 14.5. Such a lattice may not be complete. But even if it is complete the two notions of sup{M_{i} | i ∈ I} for infinite families of finitely generated submodules may differ when considered in L_{fg}(P_{R}) respectively in L_{all}(P_{R}). This phenomenon occurs in example 3.20.

(vi) If R is a ring and P_{R} a finitely generated projective right R-module, then the set L_{ds}(P_{R}) of submodules which are direct summands is a priori only a partially ordered set. Of course, if R is von Neumann regular, then L_{ds}(P_{R}) = L_{fg}(P_{R}). But for a semihereditary ring (for example a finite von Neumann algebra) we have in general only L_{ds}(P_{R}) ⊂ L_{fg}(P_{R}). The next theorem tells us that L_{ds}(A_{A}) is order isomorphic to a lattice and therefore it is a lattice. But note that it is not clear what the operations sup and inf are in terms of A-modules.

For more information on lattices and further examples see [81, Chapter III].

Proposition 3.2. Given a finite von Neumann algebra A and its algebra of affiliated operators U, all partially ordered sets in the following commutative diagram are complete lattices, and all maps are order isomorphisms and therefore lattice isomorphisms.

[figure: L_{Hilb}(l^{2}(A)); L_{P roj}(A); L_{ds}(A_{A}); L_{P roj}(U); L_{fg}(U_{U})]

The maps are given as follows:

\begin{matrix}L_{P roj}(A) → L_{Hilb}(l^{2}(A)) & p ↦ im(p : l^{2}(A) → l^{2}(A)) \\ L_{P roj}(A) → L_{P roj}(U) & p ↦ p \\ L_{P roj}(A) → L_{ds}(A_{A}) & p ↦ pA \\ L_{P roj}(A) → L_{fg}(U_{U}) & p ↦ pU \\ L_{ds}(A_{A}) → L_{fg}(U_{U}) & I ↦ IU. \end{matrix}

Here IU is the right U-module generated by I in U.

Proof. Commutativity of the diagram is obvious. That all lattices are complete follows once we have proven that they are all isomorphic from the completeness of L_{P roj}(A). In order to prove that a map is a lattice isomorphism it is sufficient to show that it is an order isomorphism. That sending p to im(p : l^{2}(A) → l^{2}(A)) yields a bijection L_{P roj}(A) → L_{Hilb}(l^{2}(A)) is well known. One verifies that this bijection is an order isomorphism. By commutativity of the square it is sufficient to deal only with three maps in the square. Since projections in U are bounded operators they already lie in A, so the lattices L_{P roj}(A) and L_{P roj}(U) coincide. Given a finitely generated right ideal I_{U} in the ∗-regular ring U there is a unique projection p ∈ U such that pU = I_{U}, compare 12.5. This leads to the bijection L_{P roj}(U) → L_{fg}(U_{U}). If p ≤ q, then multiplying pU ⊂ U from the left by q leads to pU ⊂ qU. Note that in general an order preserving bijection need not be an order isomorphism. But of course if pU ⊂ qU, then multiplying from the left by 1 − q yields p ≤ q. It remains to prove that L_{P roj}(A) → L_{ds}(A_{A}) is surjective; then injectivity follows from the commutativity of the diagram. So given a right ideal I in A which is a direct summand, there is an idempotent e such that eA = I. We have to replace the idempotent by a projection. The following lemma finishes the proof. That the lemma applies is the content of 11.2.

Lemma 3.3. In a ∗-ring R where every element of the form 1 + a^{∗}a is invertible the following holds: Given an idempotent e there always exists a projection p, such that pR = eR.

Proof. Set z = 1 − (e^{∗} − e)^{2} = 1 + (e^{∗} − e)^{∗}(e^{∗} − e). Then z = z^{∗} and ze = ez = ee^{∗}e and also z^{−1}e = ez^{−1}. If we now set p = ee^{∗}z^{−1} then p is a projection and pe = e and ep = p. This leads to pR = eR.

We are not primarily interested in these lattices of submodules, but rather in the set of isomorphism classes of such modules. This is the first step in passing from embedded submodules to abstract finitely generated projective modules and then later to arbitrary modules. The point is that over a unit-regular ring isomorphism of submodules can be expressed in lattice theoretic terms. Note that a direct sum decomposition P = M ⊕ N can be characterized by inf{M, N} = 0 and sup{M, N} = P.

Lemma 3.4. Let R be a unit-regular ring and let R_{R} be the ring considered as a right R-module. Two finitely generated submodules L and M are isomorphic if and only if they have a common complement in R_{R}, i.e. a submodule N exists with R_{R} = M ⊕ N and R_{R} = L ⊕ N.

Proof. Compare Corollary 4.4 and Theorem 4.5 in [30].

We obtain the following refined information on the diagram in Proposition 3.2.

Proposition 3.5. The lattice isomorphisms in Proposition 3.2 induce bijections of isomorphism classes, where isomorphism of projections p ≅ q means there exist elements x and y in A respectively U such that p = xy and q = yx.

Proof. Again we only have to deal with four of the five maps. The statement for the map L_{ds}(A_{A}) → L_{fg}(U_{U}) will follow from the commutativity of the square. We begin with the maps L_{P roj}(A) → L_{ds}(A_{A}) and L_{P roj}(U) → L_{fg}(U_{U}). Let R be an arbitrary ring. If p = xy and q = yx, then left multiplication by x respectively y yield mutually inverse homomorphisms between pR and qR. On the other hand given such mutually inverse homomorphisms the image of p respectively q under these homomorphisms are possible choices for x and y. Next we handle the map L_{P roj}(A) → L_{P roj}(U). The only difficulty is to show that if p and q are isomorphic (alias algebraically equivalent) inside U, then they are already isomorphic in A. The converse is obviously true. From Lemma 3.4 above we know that isomorphic finitely generated ideals in U have a common complement. We have thus expressed isomorphism in lattice theoretic terms. Since we already know from Proposition 3.2 that the map is a lattice isomorphism, p and q have a common complement in L_{P roj}(A). The following lemma, which is due to Kaplansky, finishes the proof for the map L_{P roj}(A) → L_{P roj}(U) since partial isometries are bounded and therefore in A. The lemma also tells us that projections p and q in A are isomorphic or algebraically equivalent if and only if they are Murray von Neumann equivalent. Remains to be examined the map L_{P roj}(A) → L_{Hilb}(l^{2}(A)). That isomorphic affiliated subspaces of l^{2}(A) correspond to Murray von Neumann equivalent projections was the original motivation for the definition of this notion of equivalence. On the one hand the partial isometry gives an isomorphism between affiliated subspaces. On the other hand one has to replace an arbitrary isomorphism by an isometry using the polar decomposition.

Lemma 3.6. If two projections p and q in a von Neumann algebra A have a common complement in the lattice of projections L_{P roj}(A), then they are already Murray von Neumann equivalent, i.e. there is a partial isometry u ∈ A such that p = u^{∗}u and q = uu^{∗}.

Proof. See [42, Theorem 6.6(b)]. There it is proven more generally for AW^{∗}- algebras.

Since M_{n}(A) is again a finite von Neumann algebra and its algebra of affiliated operators is isomorphic to M_{n}(U) one can apply the above results to matrix algebras.

Corollary 3.7. There is a commutative diagram of complete lattices and lattice isomorphisms, where all the maps are compatible with the different notions of isomorphism for the elements of the lattices.

[figure: L_{Hilb}(l^{2}(A)^{n}); L_{P roj}(M_{n}(A)); L_{ds}(M_{n}(A)_{M_{n}(A)}); L_{ds}(A_{A}^{n}); L_{P roj}(M_{n}(U)); L_{fg}(M_{n}(U)_{M_{n}(U)}); L_{fg}(U_{U}^{n})]

There are stabilization maps

L_{Hilb}(l^{2}(A)^{n}) → L_{Hilb}(l^{2}(A)^{n+1})
L_{P roj}(M_{n}(A)) → L_{P roj}(M_{n+1}(A))
…
L_{fg}(U_{U}^{n}) → L_{fg}(U_{U}^{n+1})

and these maps are compatible with the above lattice isomorphisms and the different notions of isomorphism for the elements of the lattices.

Proof. The map L_{ds}(M_{n}(A)_{M_{n}(A)}) → L_{ds}(A_{A}^{n}) is given by − ⊗_{M_{n}(A)} A_{A}^{n} followed by the map induced from the natural isomorphism of right A modules M_{n}(A) ⊗_{M_{n}(A)} A_{A}^{n} ≅ A_{A}^{n}. Morita equivalence tells us that − ⊗_{A} A_{M_{n}(A)}^{n} followed by a natural isomorphism is an inverse of this map. The same argument applies to U. The vertical map on the right is given by mapping a submodule M ⊂ A^{n} to the U-module it generates inside U^{n}. Since the diagram commutes, this map is also a lattice isomorphism.

For a ring R let Proj(R) denote the abelian monoid of isomorphism classes of finitely generated projective modules. An immediate consequence of the above is the following.

Theorem 3.8. The functor − ⊗_{A} U induces an isomorphism of monoids

Proj(A) → Proj(U).

In particular the natural map

K_{0}(A) → K_{0}(U)

is an isomorphism.

Proof. The maps L_{ds}(A_{A}^{n}) → L_{fg}(U_{U}^{n}) are compatible with isomorphism, stabilization and direct sums.

For more on the algebraic K-theory of A and U the reader should consult Section 10.

3.2 A Notion of Dimension for U-modules

Let us now look at dimension functions. Given a faithful normal trace tr_{A} : A → ℂ we obtain a map

dim : L_{P roj}(A) → ℝ p ↦ tr_{A}(p).

Extending the trace to matrices by

tr_{M_{n}(A)}((a_{i,j})) = ∑ tr_{A}(a_{i,i})

yields maps

dim : L_{P roj}(M_{n}(A)) → ℝ

which are compatible with the stabilization maps. Of course we normalize the traces such that tr_{M_{n}(A)}(1_{n}) = n. Because of the trace property dim is welldefined on isomorphism classes of projections. Given a finitely generated projective module over A or U there is always an isomorphic module M which is a direct summand in A^{n} respectively U^{n} for some n ∈ ℕ. Similarly a finitely generated Hilbert A-module is isomorphic to a closed submodule of l^{2}(A)^{n} for some n ∈ ℕ. Sending these modules through the diagram of 3.7 to their corresponding projections in M_{n}(A) and taking the trace gives a real number: The dimension of M. Of course for Hilbert A-modules this is the dimension considered for example in [14], [10], [11]. For an overview of possible applications see [54]. For A-modules we get the notion of dimension considered in [51]. The following proposition formulates the corresponding result for finitely generated projective U-modules.

Proposition 3.9. There is a well-defined additive real valued notion of dimension for finitely generated projective U-modules. More precisely: Given a finitely generated projective U-module M we can assign to it a real number dim_{U}(M), such that:

(i) dim_{U}(M) depends only on the isomorphism class of M.

(ii) dim_{U}(M ⊕ N) = dim_{U}(M) + dim_{U}(N).

(iii) dim_{U}(M ⊗_{A} U) = dim_{A}(M) if M is a finitely generated projective Amodule.

(iv) M = 0 if and only if dim_{U}(M) = 0.

Proof. (i) and (iii) follow immediately from 3.7. Up to isomorphism and stabilization a direct sum of modules corresponds to the block diagonal sum of projections, this yields (ii). Faithfulness of the trace implies (iv).

So far we have not used the fact that the lattices are complete. We will see that this will enable us to extend the notion of dimension to arbitrary Umodules. The following definition is completely analoguous to the definition of the dimension for A-modules given in [52].

Definition 3.10. Let M be an arbitrary U-module. Define dim′_{U}(M) ∈ [0, ∞] as

dim′_{U}(M) = sup{dim_{U}(P) | P ⊂ M, P fin. gen. projective submodule}.

The next lemma is the main technical point in proving that this dimension is well-behaved and of course it uses the completeness of the lattices. If K is a submodule of the finitely generated projective module M we define

\overline{K} = ⋂_{K⊂Q⊂M} Q ⊂ M,

where the intersection is over all finitely generated submodules Q of M, which contain K.

Lemma 3.11. Let K be a submodule of U^{n}. Since the lattice L_{fg}(U_{U}^{n}) is complete the supremum of the set {P | P ⊂ K, P finitely generated} exists.

(i) We have

\overline{K} = sup\{P | P ⊂ K, P finitely generated\}

and this module is finitely generated and therefore projective.

(ii) We have

dim′_{U}(K) = dim_{U}(\overline{K}).

Proof. (i) Let {P_{i} | i ∈ I} be the system of finitely generated submodules and {Q_{j} | j ∈ J} be the system of finitely generated modules containing K. Since every element of K generates a finitely generated submodule of K we know that K ⊂ sup{P_{i} | i ∈ I}. Since the lattice is complete sup{P_{i} | i ∈ I} is one of the finitely generated modules containing K in the definition of \overline{K}. We get \overline{K} ⊂ sup\{P_{i} | i ∈ I\}. Since P_{i} ⊂ Q_{j} for i, j arbitrary it follows that sup\{P_{i} | i ∈ I\} ⊂ Q_{j} for all j ∈ J and therefore sup\{P_{i} | i ∈ I\} ⊂ \overline{K}. (ii) From (i) we know that \overline{K} is finitely generated projective. Let p be the projection corresponding to \overline{K} and p_{i} be those corresponding to the P_{i}, then p is the limit of the increasing net p_{i} and normality of the trace implies the result.

Theorem 3.12. Let dim_{A} be the dimension for A-modules considered in [52]. The dimension dim′_{U} defined above has the following properties.

(i) Invariance under isomorphisms: dim′_{U}(M) depends only on the isomorphism class of M.

(ii) Extension: If Q is a finitely generated projective module, then

dim′_{U}(Q) = dim_{U}(Q).

(iii) Additivity: Given an exact sequence of modules

0 → M_{0} → M_{1} → M_{2} → 0

we have

dim′_{U}(M_{1}) = dim′_{U}(M_{0}) + dim′_{U}(M_{2}).

(iv) Cofinality: Let M = ⋃_{i∈I} M_{i} be a directed union of submodules (i.e.given i, j ∈ I there always exists an k ∈ I, such that M_{i}, M_{j} ⊂ M_{k}) then

dim′_{U}(M) = sup{dim′_{U}(M_{i}) | i ∈ I}.

These four properties determine dim′_{U} uniquely. Moreover the following holds.

(v) If M is an A-module, then dim′_{U}(M ⊗_{A} U) = dim_{A}(M).

(vi) If M is finitely generated projective, then dim′_{U}(M) = 0 if and only if M = 0.

(vii) Monotony: M ⊂ N implies dim′_{U}(M) ≤ dim′_{U}(N).

Notation 3.13. After having established the proof we will write dim_{U} instead of dim′_{U}. This is justified by (ii).

Proof. (i) Invariance under isomorphisms: This follows from the definition and the corresponding property 3.9 (i) for finitely generated projective modules. (ii) Extension property: Let P be a finitely generated projective submodule of the finitely generated projective module Q, then since U is von Neumann regular P is a direct summand of Q. The additivity for finitely generated projective modules 3.9 (ii) implies that dim_{U}(P) ≤ dim_{U}(Q). The claim follows. (iv) Cofinality: Let M = ⋃_{i∈I} M_{i} be a directed union. It is obvious from the definition that dim′_{U}(M_{i}) ≤ dim′_{U}(M) and therefore sup{dim′_{U}(M_{i}) | i ∈ I} ≤ dim′_{U}(M). Let now P ⊂ M be finitely generated projective. Since the system is directed there is an i ∈ I such that M_{i} contains all generators of P and therefore P itself. It follows that dim_{U}(P) ≤ dim′_{U}(M_{i}) ≤ sup{dim′_{U}(M_{i}) | i ∈ I} and finally

sup{dim_{U}(P) | P ⊂ M, P fin. gen. projective} ≤ sup{dim′_{U}(M_{i}) | i ∈ I}.

(vii) Monotony: M ⊂ N implies dim′_{U}(M) ≤ dim′_{U}(N), since on the left one has to take the supremum over a smaller set of numbers. (iii) Additivity: Let

0 → M_{0} ^{i}^{}M_{1} ^{p}^{}M_{2} → 0

be an exact sequence. For every finitely generated projective submodule P ⊂ M_{2} there is an induced sequence

0 → M_{0} → p^{−1}(P) → P → 0

which splits. So p^{−1}(P) ≅ M_{0} ⊕ P. From

{Q ⊕ P | Q ⊂ M_{0}, Q f.g.proj.} ⊂ {Q′ | Q′ ⊂ M_{0} ⊕ P, Q′ f.g.proj. }

it follows that dim′_{U}(M_{0})+dim_{U}(P) ≤ dim′_{U}(M_{0}⊕P). The monotony implies dim′_{U}(M_{0}) + dim_{U}(P) ≤ dim′_{U}(p^{−1}(P)) ≤ dim′_{U}(M_{1}). Taking the supremum over all P leads to dim′_{U}(M_{0}) + dim′_{U}(M_{2}) ≤ dim′_{U}(M_{1}). Now to the reverse inequality: Let Q ⊂ M_{1} be a finitely generated projective submodule. There are two exact sequences:

0 → i(M_{0}) ∩ Q → Q → p(Q) → 0
0 → \overline{i(M_{0}) ∩ Q} → Q → Q/\overline{i(M_{0}) ∩ Q} → 0

Because of 3.11 (i) the second one is an exact sequence of finitely generated projective modules and hence by 3.9 we already know that

dim′_{U}(Q) = dim′_{U}(\overline{i(M_{0}) ∩ Q}) + dim′_{U}(Q/\overline{i(M_{0}) ∩ Q}).

Using 3.11 (ii)

dim′_{U}(Q) = dim′_{U}(i(M_{0}) ∩ Q) + dim′_{U}(Q/\overline{i(M_{0}) ∩ Q})

follows. Now there is a split epimorphism p(Q) → Q/\overline{i(M_{0}) ∩ Q} yielding dim′_{U}(p(Q)) ≥ dim′_{U}(Q/\overline{i(M_{0}) ∩ Q}) by monotony. So

dim′_{U}(Q) ≤ dim′_{U}(i(M_{0}) ∩ Q)) + dim′_{U}(p(Q))

and monotony gives

dim′_{U}(Q) ≤ dim′_{U}(M_{0}) + dim′_{U}(M_{2}).

Taking the supremum over all finitely generated submodules Q ⊂ M_{1} finally gives dim′_{U}(M_{1}) ≤ dim′_{U}(M_{0}) + dim′_{U}(M_{2}). (vi) Was already proven in 3.9. Let us now prove uniqueness. This is done in several steps.

Step1: The extension property determines dim′_{U} uniquely on finitely generated projective modules. Step2: Let now K ⊂ Q be a submodule of a finitely generated projective module. The module K is the directed union of its finitely generated submodules K = ⋃_{i∈I} K_{i}. Since U is von Neumann regular the K_{i} are projective (semihereditary would be sufficient here). Now dim′_{U}(K) is uniquely determined by cofinality and Step1. Step3: If M is finitely generated there is an exact sequence 0 → K → U^{n} → M → 0. Additivity together with Step2 implies the result for finitely generated modules. Step4: An arbitrary module is the directed union of its finitely generated submodules and again one applies cofinality.

(v) The proof follows the same pattern as the proof of uniqueness. Note that it is shown in [52] that the dimension dim_{A} for A-modules also has the properties (i) to (iv). Step1: For finitely generated projective A-modules this is the content of 3.9(iii). Step2: A submodule K of a finitely generated projective A-module is the directed union of its finitely generated submodules K_{i} which are projective since A is semihereditary. Since −⊗_{A} U is exact and commutes with colimits K ⊗_{A} U is the directed union of the K_{i} ⊗_{A} U. Now apply cofinality of dim_{A} and dim′_{U} and use Step1. Step3: For a finitely generated module M applying − ⊗_{A} U to the exact sequence 0 → K → A^{n} → M → 0 yields an exact sequence. Now use Step2 and the additivity of dim_{A} respectively dim′_{U}. Step4: An arbitrary module is the directed union of its finitely generated submodules. Proceed as in Step2 and use Step3.

Note that this proof is independent of that one for dim_{A} in [52] which uses properties of the functor ν constructed in [51]. One could therefore take dim_{A}(M) = dim_{U}(M ⊗_{A} U) as a definition.

3.3 The Passage from A-modules to U-modules

Thinking of U as a localization of A it is no surprise that on the one hand we lose information by passing to U-modules, but on the other hand U-modules have better properties. For example, every finitely presented U-module is finitely generated projective and the category of finitely generated projective U-modules is abelian. We will now investigate this passage systematically.

Definition 3.14. For an A-module M we define its torsion submodule tM as

tM = ker(M → M ⊗_{A} U).

A module M is called a torsion module if M ⊗_{A} U = 0 or equivalently tM = M. A module is called torsionfree if tM = 0.

This is consistent with the terminology for example in [81, page 57] because U is isomorphic to the classical ring of fractions of A. An element m ∈ M lies in tM if and only if it is a torsion element in the following sense: There exists a non-zerodivisor s ∈ A, such that ms = 0. Compare [81, Chapter II, Corollary 3.3]. The module M/tM is torsionfree since \overline{m}s = \overline{ms} = 0 ∈ M/tM implies the existence of s′ with mss′ = 0 and therefore m ∈ tM. On the other hand, following [52, page 146] we make the following definition.

Definition 3.15. Let M be an A-module, then

TM = ⋃ N,

where the union is over all N ⊂ M with dim_{A}(N) = dim_{U}(N ⊗_{A} U) = 0. We denote by PM the cokernel of the inclusion TM ⊂ M.

This is indeed a submodule because for two submodules N, N′ ⊂ M with dim_{A}(N) = dim_{A}(N′) = 0 the additivity of the dimension together with

N + N′/N ≅ N′/N′ ∩ N

implies dim_{A}(N + N′) = 0. Note that dim_{A}(TM) = 0 by cofinality and TM is the largest submodule with vanishing dimension.

Note 3.16. Both t and T are left exact functors.

Proof. Under a module homomorphism torsion elements are mapped to torsion elements and a homomorphic image of a zero-dimensional module is zero-dimensional by additivity. If L ⊂ M is a submodule, then tL = L ∩ tM and TL = L ∩ TM since L ∩ TM has vanishing dimension as a submodule of TM. This implies left exactness.

We will now investigate the relation between tM and TM and the question to what degree the U-dimension is faithful. Finitely generated projective modules. If we restrict ourselves to finitely generated projective modules, we have seen in 3.8 that isomorphism classes of A- and U-modules are in bijective correspondence via −⊗_{A} U, and by adding a suitable complement one verifies that the natural map M → M ⊗_{A} U is injective. A finitely generated projective module over A or U is trivial if and only if its dimension vanishes by 3.9. Therefore tM = TM = 0 for finitely generated projective A-modules. Finitely presented modules. Let us now consider finitely presented Amodules. This category of modules was investigated in [51]. Since the ring A is semihereditary it is an abelian category. (This is more generally true iff the ring is coherent.) Recall that Proj(R) denotes the monoid of isomorphism classes of finitely generated projective modules over R.

Proposition 3.17. Let M be a finitely presented A-module. Then

(i) M ⊗_{A} U is finitely generated projective.

(ii) TM = tM.

(iii) M is a torsion module if and only if dim_{A}(M) = 0. Every torsion module is the cokernel of a weak isomorphism.

(iv) PM = M/TM is projective and M ≅ PM ⊕ TM.

(v) Under the isomorphism Proj(A) → Proj(U), respectively K_{0}(A) → K_{0}(U) the class [PM] corresponds to [M ⊗_{A} U].

Proof. (i) By right exactness of the tensor product M ⊗_{A} U is finitely presented. Over the von Neumann regular ring U this implies being finitely generated projective. (iv) is proven in [51]. Now M ⊗_{A} U ≅ PM ⊗_{A} U ⊕ TM ⊗_{A} U ≅ PM because TM ≅ coker(f) for some weak isomorphism f : A^{n} → A^{n} and by 2.6 and right exactness of the tensor product we get TM ⊗_{A} U ≅ coker(f) ⊗_{A} U ≅ coker(f ⊗_{A} id_{U}) ≅ 0. The rest follows.

Arbitrary modules. In general tM and TM differ. Counterexamples can already be realized by finitely generated modules. More precisely the following holds.

Proposition 3.18. Let M be an A-module.

(i) tM ⊂ TM and torsion modules have vanishing dimensions.

(ii) There exists a finitely generated A-module with tM = 0 and TM = M.

(iii) M is a torsion module if and only if it is the directed union of quotients of finitely presented torsion modules.

(iv) If M is finitely generated PM = M/TM is projective and M ≅ PM ⊕ TM.

(v) There exists a finitely generated module M with

dim_{A}(M) = dim_{U}(M ⊗_{A} U) = 0

but M ⊗_{A} U ≠ 0.

Proof. (i) It suffices to show that dim_{A}(tM) = dim_{U}(tM ⊗_{A}U) = 0. Now tM consists of torsion elements and therefore tM ⊗_{A} U = 0. (v) follows from (ii) and we will give an example of such a module below. (iv) was proven in [52]. Let us prove (iii): A quotient of a torsion module is a torsion module, and a directed union of torsion modules is again a torsion module. On the other hand suppose M is a torsion module. M = ⋃_{i∈I} M_{i} is the directed union of its finitely generated submodules. Since any submodule of a torsion module is a torsion module it remains to be shown that a finitely generated torsion module N is always a quotient of a finitely presented torsion module. Choose a surjection p : A^{n} → N and let K = ker(p) be the kernel. Since N ≅ A^{n}/K is a torsion module, for every a ∈ A^{n} there exists a non-zerodivisor s ∈ A such that as ∈ K. Let e_{i} = (0, …, 1, …, 0)^{tr} be the standard basis for A^{n} and choose s_{i} with e_{i}s_{i} ∈ K. Note that

diag(0, …, s_{i}, …, 0)e_{i} = e_{i}s_{i}.

Here diag(b_{1}, …, b_{n}) denotes the diagonal matrix with the corresponding entries. Since K is a right A-module we have for an arbitrary vector (a_{1}, …, a_{n})^{tr} in A^{n} that

diag(s_{1}, …, s_{n})(a_{1}, …, a_{n})^{tr} = ∑ e_{i}s_{i}a_{i} ∈ K.

Let S denote the right linear map A^{n} → A^{n} corresponding to the diagonal matrix diag(s_{1}, …, s_{n}), then im(S) ⊂ K and N ≅ A^{n}/K is a quotient of the finitely presented module A^{n}/im(S). Moreover A^{n}/im(S) ≅ ⨁ A/s_{i}A is a torsion module by Proposition 3.17(iii) and 2.6.

Note 3.19. In [55, Definition 2.1] we defined a module to be cofinal-measurable if all its finitely generated submodules are quotients of finitely presented zero-dimensional modules. By Proposition 3.17(iii) and 3.18(iii) we see that a module is cofinal-measurable if and only if it is a torsion module.

Example 3.20. We will now give two counterexamples to finish the proof of the above proposition. Both examples follow the same pattern. Let I_{i}, i ∈ I be a directed family of right ideals in A such that each I_{i} is a direct summand, dim_{A}(I_{i}) < 1 and sup_{i∈I}(dim_{A}(I_{i})) = 1. Note that I = ⋃_{i∈I} I_{i} ≠ A, because 1 ∈ I_{i} for some i would contradict dim_{A}(I_{i}) < 1. Since I_{i} is a direct summand A/I_{i} → (A/I_{i}) ⊗_{A} U is injective. Using that − ⊗_{A} U is exact and commutes with colimits one verifies that

A/I → (A/I) ⊗_{A} U ≅ A ⊗_{A} U/I ⊗_{A} U ≅ A ⊗_{A} U/ ⋃(I_{i} ⊗_{A} U)

is injective as well. Therefore t(A/I) = 0. On the other hand additivity and cofinality of the dimension imply dim_{A}(A/I) = dim_{U}((A/I) ⊗_{A} U) = 0. Here are two concrete examples where such a situation arises.

(i) Take A = Ł^{∞}(S^{1}, μ), the essentially bounded functions on the unit circle with respect to the normalized Haar measure μ on S^{1}. Let X_{i}, i ∈ ℕ be an increasing sequence of measurable subsets of S^{1} such that μ(X_{i}) < 1 and sup_{i∈ℕ}(μ(X_{i})) = 1. The corresponding characteristic functions χ_{X_{i}} generate ideals in A with the desired properties, since dim_{A}(χ_{X_{i}}A) = μ(X_{i}).

(ii) Let Γ be an infinite locally finite group, i.e. every finitely generated subgroup is finite. Take A as N Γ. Let I denote the set of finite subgroups. For every H ∈ I let ϵ_{H} : ℂH → ℂ denote the augmentation ∑ a_{h}h ↦ ∑ a_{h}. Since H is finite the augmentation ideal ker(ϵ_{h}) is a direct summand of ℂH. Since ℂH = N H and − ⊗_{N H} N Γ is exact and compatible with dimensions we get that ker(ϵ_{H}) ⊗_{ℂH} N Γ is a direct summand of N Γ and

dim_{N Γ}(ker(ϵ_{H}) ⊗_{ℂH} N Γ) = 1 − \frac{1}{|H|}.

The system is directed because for finite subgroups H and G with H ⊂ G we have ker(ϵ_{H}) ⊗_{ℂH} ℂG ⊂ ker(ϵ_{G}). If we tensor the exact sequence

⋃_{H} ker(ϵ_{H}) ⊗_{ℂH} ℂΓ → ℂΓ → ℂ ⊗_{ℂΓ} ℂΓ → 0

with N Γ and use compatibility with colimits and right exactness we get

H_{0}^{Γ}(EΓ; N Γ) ≅ ℂ ⊗_{ℂΓ} N Γ ≅ N Γ/ ⋃_{H} ker(ϵ_{H}) ⊗_{ℂH} N Γ.

So we can even realize the counterexample as the homology of a space.

4 New Interpretation of L^{2}-Invariants

We will now briefly summarize how the notions developed in the last section apply in order to provide alternative descriptions of L^{2}-invariants. Recall the following definitions from [52, Section 4]. Let Γ be a group. Given an arbitrary Γ-space X one defines the singular homology of X with twisted coefficients in the ℤΓ-module N Γ as

H_{p}^{Γ}(X; N Γ) = H_{p}(C_{∗}^{sing}(X) ⊗_{ℤΓ} N Γ),

where C_{∗}^{sing}(X) is the singular chain complex of X considered as a complex of right ℤΓ-modules. Here H_{p}^{Γ}(X; N Γ) is still a right N Γ-module and therefore

b_{p}^{(2)}(X) = dim_{A}(H_{p}^{Γ}(X; N Γ))

makes sense. This definition extends the definition of L^{2}-Betti numbers via Hilbert N Γ-modules for regular coverings of CW-complexes of finite type to arbitrary Γ-spaces. Completely analoguous we make the following definition.

Definition 4.1. Let X be a Γ-space, then

H_{p}^{Γ}(X; UΓ) = H_{p}(C_{∗}^{sing}(X) ⊗_{ℤΓ} UΓ).

Proposition 4.2. Let X be an arbitrary Γ-space.

(i) H_{p}^{Γ}(X; N Γ) ⊗_{N Γ} UΓ ≅ H_{p}^{Γ}(X; UΓ) as UΓ-modules.

(ii) b_{p}^{(2)}(X) = dim_{U}(H_{p}^{Γ}(X; UΓ)).

(iii) tH_{p}^{Γ}(X; N Γ) = ker(H_{p}^{Γ}(X; N Γ) → H_{p}^{Γ}(X; UΓ)).

If X is a regular covering of a CW-complex of finite type, then

(iv) H_{p}^{Γ}(X; N Γ) is finitely presented, H_{p}^{Γ}(X; UΓ) is finitely generated projective and under the isomorphism Proj(N Γ) → Proj(UΓ), respectively K_{0}(N Γ) → K_{0}(UΓ) the class [PH_{p}^{Γ}(X; N Γ)] corresponds to [H_{p}^{Γ}(X; UΓ)].

(v) tH_{p}^{Γ}(X; N Γ) = TH_{p}^{Γ}(X; N Γ) = ker(H_{p}^{Γ}(X; N Γ) → H_{p}^{Γ}(X; UΓ)).

Proof. Since −⊗_{N Γ}UΓ is exact it commutes with homology. If X is a regular covering of finite type the singular chain complex is quasi-isomorphic to the cellular chain complex which consists of finitely generated free ℤΓ-modules. Since finitely presented N Γ-modules form an abelian category the homology modules H_{p}^{Γ}(X; N Γ) are again finitely presented. The rest follows from the preceding results about t, T and the dimension.

As far as dimension is concerned one can therefore work with U-modules. Tensoring with U is the algebraic analogue of the passage from unreduced to reduced L^{2}-homology in the Hilbert space set-up. If one is interested in finer invariants like the Novikov-Shubin invariants the passage to U-modules is too harsh. The torsion submodule tH_{p}^{Γ}(X; N Γ) carries the information of the Novikov-Shubin invariants in case X is a regular covering of a CWcomplex of finite type and seems to be the right candidate to carry similar information in general. Compare [55] and Note 3.19.

5 The Atiyah Conjecture

5.1 The Atiyah Conjecture

As already mentioned in the introduction one of the main motivations to study the algebras ℂΓ, DΓ, RΓ, N Γ and UΓ is the following conjecture, which goes back to a question of Atiyah in [1]. We denote by Fin(Γ) the set of finite subgroups of Γ and \frac{1}{|FinΓ|}ℤ is the additive subgroup of ℝ generated by the set of rational numbers \{\frac{1}{|H|}|H a finite subgroup of Γ\}. In particular \frac{1}{|FinΓ|}ℤ = ℤ if Γ is a torsionfree group and if there is a bound on the order of finite subgroups \frac{1}{|FinΓ|}ℤ = \frac{1}{l}ℤ, where l is the least common multiple of these orders. In general \frac{1}{|FinΓ|}ℤ ⊂ ℚ.

Conjecture 5.1 (Atiyah Conjecture). If M is a closed manifold with fundamental group Γ and universal covering M̃, then

b_{p}^{(2)}(M̃) ∈ \frac{1}{|FinΓ|}ℤ for all p ≥ 0.

There is a topological and a purely algebraic reformulation of this conjecture.

Proposition 5.2. Let Γ be a finitely presented group.The following statements are equivalent.

(i) For every finitely presented ℤΓ-module M

dim_{UΓ}(M ⊗_{ℤΓ} UΓ) ∈ \frac{1}{|FinΓ|}ℤ.

(ii) For every ℤΓ-linear map f between finitely generated free ℤΓ-modules

dim_{UΓ}(ker(f ⊗_{ℤΓ} id_{UΓ})) ∈ \frac{1}{|FinΓ|}ℤ.

(iii) If X is a connected CW-complex of finite type with fundamental group Γ, then

b_{p}^{(2)}(X̃) ∈ \frac{1}{|FinΓ|}ℤ for all p ≥ 0.

(iv) If M is a closed manifold with fundamental group Γ, then

b_{p}^{(2)}(M̃) ∈ \frac{1}{|FinΓ|}ℤ for all p ≥ 0.

Proof. Using the additivity of the dimension we verify that in the statement of (ii) one can replace the kernel of the map f ⊗_{ℤΓ} id_{UΓ} by the image or the cokernel. Since tensoring is right exact coker(f ⊗_{ℤΓ} id_{UΓ}) ≅ coker(f) ⊗_{ℤΓ} UΓ and we see that (i) and (ii) are equivalent. Let us now prove that (ii) implies (iii). The cellular chain complex (C_{∗}, d_{∗}) = (C_{∗}^{cell}(X̃), d_{∗}) of X̃ is a complex of finitely generated free right ℤΓ-modules. By definition b_{p}^{(2)}(X̃) = dim_{UΓ} H_{p}(C_{∗} ⊗_{ℤΓ} UΓ). The exact sequences

0 → ker(d_{n} ⊗ id_{UΓ}) →  C_{n} ⊗ UΓ  → im(d_{n} ⊗ id_{UΓ}) → 0
0 → im(d_{n+1} ⊗ id_{UΓ}) → ker(d_{n} ⊗ id_{UΓ}) → H_{n}(C_{∗} ⊗ UΓ) → 0

together with (ii) and additivity of dimension imply (iii). Since (iv) is a special case of (iii) it remains to be shown that (iv) implies (ii). Since Γ is finitely presented there is a finite 2-complex with fundamental group Γ. Now attach m 3-cells with a trivial attaching map and n 4-cells in such a way that the differential d_{4} realizes f. Embed this finite CW-complex X in ℝ^{N} and take M as the boundary of a regular neighbourhood. There is a 5-connected map M → X and therefore H_{4}(M̃, UΓ) = H_{4}(X̃, UΓ) = ker(f ⊗ id_{UΓ}).

It is convenient to formulate the algebraic version of the Atiyah conjecture not only for the integral group ring but also for group rings with other coefficients.

Conjecture 5.3 (Atiyah Conjecture with Coefficients). Let R ⊂ ℂ be a subring of the complex numbers. For every finitely presented RΓ-module M

dim_{UΓ}(M ⊗_{RΓ} UΓ) ∈ \frac{1}{|FinΓ|}ℤ.

The conjecture with coefficients R ⊂ ℂ implies the conjecture with integer coefficients. In fact the results obtained by Linnell, which we will discuss later, all hold even for complex coefficients. Conversely we have at least:

Lemma 5.4. Let R ⊂ ℂ be a subring and denote by Q(R) the subfield of ℂ generated by R. The Atiyah conjecture with R-coefficients implies the Atiyah conjecture with Q(R)-coefficients. In particular the genuine Atiyah conjecture implies the Atiyah conjecture with ℚ-coefficients.

Proof. The field Q(R) is the localization RS^{−1}, where S = R − {0}. Given a map f : RΓ^{n} → RΓ^{m} we interpret it as a matrix and multiply by a suitable s ∈ S to clear denominators. This does not change the dimension of the kernel or the image of f⊗id_{UΓ} since multiplication by s is an isomorphism.

Another easy observation is that the conjecture is stable under directed unions.

Lemma 5.5. Suppose Γ = ⋃_{i∈I} Γ_{i} is the directed union of the subgroups Γ_{i} and the Atiyah conjecture with R-coefficients holds for each Γ_{i}, then it also holds for Γ.

Proof. Every entry of a matrix representing f : RΓ^{n} → RΓ^{m} involves only finitely many group elements. Since the union is directed we can find an i ∈ I such that f is a matrix over RΓ_{i}. Now

dim_{UΓ}(coker(f ⊗ id_{UΓ})) = dim_{UΓ}(coker(f ⊗ id_{UΓ_{i}})) ⊗_{UΓ_{i}} UΓ

= dim_{UΓ_{i}}(coker(f ⊗ id_{UΓ_{i}})) ∈ \frac{1}{|FinΓ_{i}|}ℤ ⊂ \frac{1}{|FinΓ|}ℤ

since the dimension behaves well with respect to induction and tensoring is right exact.

Similarly, at least for torsionfree groups, the conjecture is stable under passage to subgroups. The Atiyah conjecture is related to the zero divisor conjecture.

Conjecture 5.6 (Zero Divisor Conjecture). The rational group ring ℚΓ of a torsion free group contains no zero divisors.

The converse of this statement is true, since given a finite subgroup H ⊂ Γ the norm element \frac{1}{|H|} ∑_{h∈H} h is an idempotent and therefore a zerodivisor. Of course this conjecture could also be stated with other coefficients.

Note 5.7. The Atiyah conjecture implies the zero divisor conjecture.

Proof. Suppose a ∈ ℚΓ is a zerodivisor. Left multiplication with a is a ℤΓlinear map and therefore dim_{UΓ} ker(a ⊗ id_{UΓ}) ∈ ℤ by the Atiyah conjecture with rational coefficients. Since ker(a⊗id_{UΓ}) is a submodule of ℤΓ⊗_{ℤΓ}UΓ ≅ UΓ, we have that dim_{UΓ}(ker(a ⊗ id_{UΓ})) is either 0 or 1. In the second case a = 0. In the first case ker a ⊂ ker(a ⊗ id_{UΓ}) = 0 because ker(a ⊗ id_{UΓ}) is a finitely generated projective module and such a module vanishes if and only if its dimension is zero.

We will briefly discuss how the Atiyah conjecture is related to questions about Euler characteristics of groups. If a group Γ has a finite model for its classifying space BΓ, then its Euler characteristic χ(Γ) = χ(BΓ) is defined. Note that such a group is necessarily torsionfree. If Γ′ ⊂ Γ is a subgroup of finite index we have

χ(Γ′) = [Γ : Γ′] · χ(Γ).

This is the fundamental property which allows to extend the definition to not necessarily torsionfree groups. A group Γ is said to have virtually a finite classifying space if a subgroup Γ′ of finite index admits a finite classifying space. In that case we define

χ_{virt}(Γ) = \frac{1}{[Γ : Γ′]}χ(Γ′).

This idea goes back to Wall [84]. It is possible to extend the definition further to groups Γ which are of finite homological type. For the definition of groups of finite homological type and more details we refer to Capter IX in [8]. The following Theorem was conjectured by Serre and proven by Brown [7].

Theorem 5.8 (Serre’s Conjecture). If Γ is of finite homological type, then

χ_{virt}(Γ) ∈ \frac{1}{|FinΓ|}ℤ.

Proof. See Theorem 9.3 on page 257 in [8] and page 246 in the same book for the definition of finite homological type.

Now if we consider the passage to a subgroup of finite index the L^{2}-Betti numbers behave like the Euler characteristic. Namely let Y be a Γ-space and Γ′ be a subgroup of finite index, then

b_{p}^{(2)}(Y; Γ) = \frac{1}{[Γ : Γ′]}b_{p}^{(2)}(Y; Γ′).

Therefore the L^{2}-Euler characteristic

χ^{(2)}(Γ) = ∑_{p=0}^{∞}(−1)^{p}b_{p}^{(2)}(EΓ),

which is known to coincide with the ordinary Euler characteristic in the case of a finite complex, coincides with the virtual Euler characteristic if Γ virtually admits a finite classifying space. For more details we refer to [11] and [53].

Note 5.9. The Atiyah conjecture implies

χ^{(2)}(Γ) ∈ \frac{1}{|FinΓ|}ℤ.

Therefore the theorem of Brown above gives further evidence for the Atiyah conjecture.

5.2 A Strategy for the Proof

The main objective of this section is the following strategy for a proof of the above mentioned conjectures.

Theorem 5.10. Let Γ be a discrete group. Suppose there is an intermediate ring ℂΓ ⊂ SΓ ⊂ UΓ such that

(A) The ring SΓ is von Neumann regular.

(B) The map i_{Γ} induced by the natural induction maps

i_{Γ} : ⨁_{K∈FinΓ} K_{0}(ℂK) → K_{0}(SΓ)

is surjective.

Then the Atiyah conjecture with complex coefficients is true for the group Γ.

Proof. Suppose M is a finitely presented ℂΓ-module. It does not define a class in K_{0}(ℂΓ), but since every finitely presented module over the von Neumann regular ring SΓ is projective, M ⊗_{ℂΓ}SΓ defines a class in K_{0}(SΓ). Now the surjectivity of i_{Γ} implies that there is an element ([N_{K}] − [L_{K}])_{K} ∈ ⨁_{K∈FinΓ} K_{0}(ℂK) which maps to [M ⊗_{ℂΓ} SΓ]. Here N_{K} and L_{K} are finitely generated projective ℂK-modules. Since the Γ-dimension behaves well with respect to induction, and for finite groups it coincides with the complex dimension divided by the group order, we have

dim_{UΓ}(ker(f ⊗ id_{SΓ}) ⊗ UΓ) = ∑_{K} dim_{UΓ}([N_{K} ⊗ UΓ] − [L_{K} ⊗ UΓ])
= ∑_{K} dim_{ℂK}(N_{K}) − dim_{ℂK}(L_{K})
= ∑_{K} \frac{1}{|K|} (dim_{ℂ}(N_{K}) − dim_{ℂ}(L_{K})) ,

which is obviously in the subgroup \frac{1}{|FinΓ|}ℤ.

Roughly speaking condition (A) says that we have to make our ring large enough to have good ring theoretical properties (remember that at least UΓ itself is von Neumann regular). Condition (B) says we cannot make it too big because otherwise its K-theory becomes too large. Here are some remarks on the proof.

(i) The same proof applies if we replace the set FinΓ of finite subgroups in condition (B) by any set F(Γ) of subgroups for which the Atiyah conjecture is already known, because then we have for K ∈ F(Γ) and [N_{K}] ∈ K_{0}(ℂK) that dim_{UK}(N_{K}) ∈ \frac{1}{|FinK|}ℤ ⊂ \frac{1}{|FinΓ|}ℤ. For example one could take the virtually cyclic subgroups of Γ.

(ii) Of course we can formulate a similar statement for other than complex coefficients. This might be an advantage for verifying (B), compare the next section.

(iii) Note that we actually proved that (A) and (B) implies

im(K_{0}(SΓ) → K_{0}(UΓ) ^{dim}^{}ℝ) = \frac{1}{|FinΓ|}ℤ.

(iv) Working with the kernel instead of the cokernel of a map f between finitely generated free ℂΓ-modules one verifies that instead of von Neumann regularity it would be sufficient that the ring SΓ is semihereditary and UΓ is flat as an SΓ-module. Remember that a ring is semihereditary if every finitely generated submodule of a projective module is projective. This would imply that im(f ⊗ id_{SΓ}) and therefore also ker(f ⊗ id_{SΓ}) is finitely generated projective. In fact it would be sufficient to ensure that one of these modules has a finite resolution by finitely generated projective modules.

(v) It seems that every reasonable construction of a ring SΓ leads to a ring which is closed under the ∗-operation. We therefore expect SΓ to be a ∗-closed subring of UΓ.

Later, when we discuss Linnell’s results on the Atiyah conjecture, we will actually find an SΓ which is not only von Neumann regular but semisimple. Nevertheless we believe that in general the above should be the right formulation for the following reasons.

Example 5.11. Let Γ be an infinite locally finite group (every finitely generated subgroup is finite), as for example Γ′ = ⨁_{n∈ℕ} ℤ/nℤ. Then Γ = ⋃_{i∈I} Γ_{i} is the directed union of its finite subgroups. Using the elementwise criterion for von Neumann regularity (compare 12.1(i)) one verifies that ℂΓ = ⋃_{i∈I} ℂΓ_{i} is von Neumann regular. Also (B) holds. Therefore we conclude from the above remark that im(K_{0}(ℂΓ) → K_{0}(UΓ) → ℝ) = \frac{1}{|FinΓ|}ℤ. This group is not finitely generated (for Γ′ it is ℚ). Suppose SΓ is a semisimple intermediate ring, then K_{0}(SΓ) is finitely generated and the above map to ℝ factorizes over K_{0}(SΓ). A contradiction.

We see that in general we cannot expect a semisimple intermediate ring SΓ. On the other hand, when \frac{1}{|FinΓ|}ℤ is a discrete subgroup of ℝ semisimplicity already follows from (A) and (B):

Proposition 5.12. Suppose the intermediate ring SΓ fulfills the conditions (A) and (B) and there is a bound on the orders of finite subgroups of Γ, then SΓ is semisimple.

Proof. By assumption we know that

im(K_{0}(SΓ) → K_{0}(UΓ) → ℝ) = \frac{1}{|FinΓ|}ℤ = \frac{1}{l}ℤ

for a positive integer l. We will see that there can be no strictly increasing chain I_{1} ⊂ I_{2} ⊂ … ⊂ I_{r} ⊂ SΓ of right ideals with r > l. With 12.3(iii) this implies the result. Choose x_{i} ∈ I_{i} \ I_{i−1} and let J_{i} be the ideal generated by x_{1}, … x_{i}, then the J_{i} form a strictly increasing chain of finitely generated ideals of the same length. Since SΓ is von Neumann regular J_{r} is a direct summand of SΓ and J_{i−1} is a direct summand of J_{i}. We see that dim_{UΓ}(J_{i−1}⊗ UΓ) is strictly smaller than dim_{UΓ}(J_{i} ⊗ UΓ). Here we are using the fact that for finitely generated projective SΓ-modules the dimension function dim_{UΓ}(− ⊗ UΓ) is faithful. Simply represent a finitely generated projective module P over SΓ by an idempotent e ∈ M(SΓ) and verify that the same idempotent considered as a matrix over M(UΓ) represents P ⊗_{SΓ} UΓ. Now since dim_{UΓ} is faithful dim_{UΓ}(P ⊗ UΓ) = 0 implies that P ⊗ UΓ = 0 and therefore also the idempotent e is trivial. Now dim_{UΓ}(J_{i} ⊗ UΓ) ∈ [0, 1] ∩ \frac{1}{l}ℤ since J_{i} ⊗ UΓ is a submodule of UΓ. Therefore the chain can at most have length l.

Similarly we have for torsionfree groups:

Proposition 5.13. If Γ is torsionfree and SΓ is an intermediate ring of ℂΓ ⊂ UΓ, then conditions (A) and (B) hold for SΓ if and only if SΓ is a skew field.

Proof. Suppose SΓ fulfills (A) and (B). Since \frac{1}{|FinΓ|}ℤ = ℤ we can take l = 1 in the proof above. So we have a semisimple ring which has no nontrivial ideals. On the other hand a skew field is von Neumann regular and K_{0}(SΓ) = ℤ generated by [SΓ] if SΓ is a skew field and therefore (B) is obviously satisfied.

We have seen in the preceding section that the Atiyah conjecture is stable under directed unions. It is therefore reassuring that conditions (A) and (B) also show this behaviour. At the same time the following generalizes Example 5.11.

Proposition 5.14. Suppose Γ = ⋃_{i∈I} Γ_{i} is a directed union and there are intermediate rings SΓ_{i} with SΓ_{i} ⊂ SΓ_{j} for Γ_{i} ⊂ Γ_{j} which all fulfill the conditions (A) and (B), then the ring SΓ = ⋃_{i∈I} SΓ_{i} also fulfills (A) and (B).

Proof. A directed union of von Neumann regular rings is von Neumann regular and K-theory is compatible with colimits.

An intermediate ring SΓ is by no means uniquely determined by (A) and (B).

Example 5.15. Let Γ = ℤ be an infinite group. We know that in this case we can identify the rings ℂΓ and UΓ with the ring of Laurent polynomials ℂ [z^{±1}] and the ring Ł(S^{1}) of (classes of) measurable functions on the unit circle S^{1}, where we embed the circle in the complex plane and z denotes the standard coordinate. One possible choice for SΓ is in this case the field of rational functions ℂ (z). But one could equally well take the field of meromorphic functions of any domain of the complex plane which contains S^{1}. We see that there is a continuum of possible choices.

But in fact there are two natural candidates for the ring SΓ: the rational closure R(ℂΓ ⊂ UΓ) of ℂΓ in UΓ and the division closure D(ℂΓ ⊂ UΓ) of ℂΓ in UΓ. They constitute a kind of minimal possible choice for SΓ. We will discuss these rings in detail in Section 8 and in Appendix III. Finally we should mention the following result which is implicit in [48].

Theorem 5.16. Suppose Γ is torsionfree, then the Atiyah conjecture with coefficients in R holds for Γ if and only if the division closure D(RΓ ⊂ UΓ) of RΓ in UΓ is a skew field.

Proof. This will be proven as Proposition 8.30 in Section 8.

5.3 The Relation to the Isomorphism Conjecture in Algebraic K-theory

The map i_{Γ} in condition (B) of Theorem 5.10 factorizes as follows:

⨁_{H∈FinΓ} K_{0}(ℂH) \overline{i_{Γ}_{}}K_{0}(SΓ)

[figure: colim_{H∈FinΓ}K_{0}(ℂH); j_{Γ}; K_{0}(ℂΓ)]

The map on the left is (by construction) always surjective. In this subsection we will explain why a variant of the isomorphism conjecture in algebraic K-theory ([27], [19]) implies that the bottom map is an isomorphism. To form the colimit we understand FinΓ as a category with morphisms given by inclusions K ⊂ H and conjugation maps c_{g} : H → ^{g}H, h ↦ ghg^{−1} with g ∈ Γ. These maps induce ring homomorphisms of the corresponding group algebras and K_{0}(ℂ? ) can be considered as a functor from FinΓ to the category of abelian groups. Equivalently we could work with the orbit category: The orbit category OrΓ has as objects the homogeneous spaces Γ/H, and morphisms are Γequivariant maps. Let Or(Γ, Fin) be the full subcategory whose objects are the homogeneous spaces Γ/H with H finite. There is an obvious bijection between the set of objects of FinΓ and Or(Γ, Fin) but the morphisms differ. This difference vanishes on the level of K-theory since an inner automorphism c_{h} : H → H with h ∈ H induces the identity map on K_{0}.

colim_{FinΓ}K_{0}(ℂ? ) = colim_{Or(Γ,Fin)}K_{0}(ℂ? ).

One possible formulation of the isomorphism conjecture is as follows:

Conjecture 5.17. The assembly map

Kℂ_{∗}^{OrΓ}(E(Γ, Fin)) ^{Ass}^{}K_{∗}(ℂΓ)

is an isomorphism.

We will not explain the construction of the assembly map. The interested reader should consult [19]. But let us say a few words about the source and target of the assembly map to get an idea what it is about. The target: The target of the assembly map consists of the algebraic Kgroups K_{n}(ℂΓ), n ∈ ℤ of the group ring ℂΓ. The isomorphism conjecture should be seen as a means to compute these groups. The source: Kℂ_{∗}^{OrΓ}( −) is a generalized equivariant homology theory similar to Bredon homology (see [6]), but with more complicated coefficients. E(Γ, Fin) is the classifying space for the family of finite subgroups [23, I.6], i.e. it is a Γ-CW complex which is uniquely determined up to Γ-equivariant homotopy by the following property of its fixed point sets.

E(Γ, Fin)^{H} = \{ \begin{matrix}∅ & H ∉ FinΓ \\ contractible & H ∈ FinΓ \end{matrix} .

There is an equivariant version of an Atiyah-Hirzebruch spectral sequence [19, Section 4] which helps to actually compute the source of the assembly map.

H_{p}(C_{∗}(E(Γ, Fin)^{?})) ⊗_{ℤOrΓ} K_{q}(ℂ? )) ⇒ Kℂ_{∗}^{OrΓ}(E(Γ, Fin)).

Here C_{∗}(E(Γ, Fin)^{?}) is the cellular chain complex of the space E(Γ, Fin) viewed as a functor from the category OrΓ to the category of chain complexes by sending Γ/H to the cellular chain complex of the fixed point set. The tensor product is a tensor product over the orbit category [50, 9.12 on page 166]. In analogy to ordinary group homology one should think of the ℤOrΓchain complex as a free resolution of the ℤOrΓ-module ℤ_{Fin}( ? ) which is given by

ℤ_{Fin}(H) = \{ \begin{matrix}0 & H ∉ FinΓ \\ ℤ & H ∈ FinΓ \end{matrix} .

We see that since E(Γ, Fin)^{H} = ∅ for infinite H only the K_{∗}(ℂH) for H finite enter the computation. So the isomorphism conjecture is a device to compute K_{∗}(ℂΓ) from the knowledge of the K-theory of the finite subgroups. How this information is assembled is encoded in the space E(Γ, Fin).

Note 5.18. Suppose the isomorphism conjecture 5.17 is true. Then

(i) The negative K-groups K_{−i}(ℂΓ) vanish.

(ii) There is an isomorphism colim_{K∈FinΓ}K_{0}(ℂK) →^{≅} K_{0}(ℂΓ).

Proof. For a finite group H the group algebra ℂH is semisimple, and in particular it is a regular ring. (Here regular does not mean von Neumann regular!). It is known that the negative K-groups for such rings vanish [74, page 154]. We see that the above spectral sequence becomes a first quadrant spectral sequence and in particular K_{−i}(ℂΓ) = 0. Moreover at the origin (p, q) = (0, 0) there are no differentials and no extension problems to solve. We see that

K_{0}(ℂΓ) ≅ H_{0}(C_{∗}(E(Γ, Fin)^{?}) ⊗_{ℤOrΓ} K_{0}(ℂ? )).

Since the ℤOrΓ-chain complex C_{∗}(E(Γ, Fin)^{?}) is a free resolution of ℤ_{Fin}( ? ) and the tensor product is right exact [50, 9.23 on page 169] we get

H_{0}(C_{∗}(E(Γ, Fin)^{?}) ⊗_{ℤOrΓ} K_{0}(ℂ? )) ≅ ℤ_{FinΓ}( ? ) ⊗_{ℤOrΓ} K_{0}(ℂ? ).

From the definition of the tensor product over the orbit category we get

ℤ_{FinΓ}( ? ) ⊗_{ℤOrΓ} K_{0}(ℂ? ) = colim_{FinΓ}K_{0}(ℂ? ).

The map which gives the isomorphism in 5.18(ii) should coincide with the natural map j_{Γ} above.

Proposition 5.19. The map

j_{Γ} : colim_{K∈FinΓ}K_{0}(ℂK) → K_{0}(ℂΓ)

is always rationally injective, i.e. j_{Γ} ⊗ id_{ℚ} is injective.

Proof. The proof uses the Hattori-Stallings rank which is also known as the Bass trace map. For a ring R denote by [R, R] the subgroup of the additive group of R generated by commutators, i.e. elements of the form ab − ba. If R is a ℂ-algebra this subgroup is a complex vector space. Every additive map f out of R which has the trace property f(ab − ba) = 0 factorizes over HC_{0}(R) = R/ [R, R]. The natural map R → HC_{0}(R), a ↦ \overline{a} extends to matrices via

(a_{ij}) ↦ ∑_{i} \overline{a_{ii}}.

This extension still has the trace property and is compatible with stabilization. It leads therefore to a map

ch_{0} : K_{0}(R) → HC_{0}(R).

This is the Hattori-Stallings rank, which can also be considered as a Chern character [49, Section 8.3]. HC_{0} is functorial and ch_{0} is a natural transformation. We obtain a commutative diagram

colim_{H∈FinΓ}K_{0}(ℂH) \overline{j_{Γ}_{}}K_{0}(ℂΓ)

[figure: colim_{H∈FinΓ}HC_{0}(ℂH); f; HC_{0}(ℂΓ)]

Note that it is sufficient to show that j_{Γ} ⊗ ℂ is injective. We claim that the bottom map in the above diagram is injective and the complex linear extension f_{ℂ} of f is an isomorphism. Given a group G we denote by conG the set of conjugacy classes of G. The natural quotient map G → conG induces a map of free vector spaces ℂG → ℂconG. One checks that the kernel of this map is [ℂG, ℂG] and therefore HC_{0}(ℂG) can be naturally (!) identified with ℂconG. Since the functor free vector space is left adjoint to the forgetful functor it commutes with colimits. Therefore, in order to show that the bottom map is injective it is sufficient to show that the map

colim_{K∈FinΓ}conK → conΓ

is injective. Given two elements h and k with h = gkg^{−1} for some g ∈ Γ the conjugation map c_{g} is a map in the category FinΓ and therefore the corresponding conjugacy classes are already identified in the colimit. To show that f_{ℂ} is an isomorphism it is sufficient to show that for every finite group H the map

ch_{0}(H)_{ℂ} : K_{0}(ℂH) ⊗_{ℤ} ℂ → HC_{0}(ℂH)

is an isomorphism because

colim_{H∈FinΓ} (K_{0}(ℂH) ⊗_{ℤ} ℂ) ≅ colim_{H∈FinΓ} (K_{0}(ℂH)) ⊗_{ℤ} ℂ.

A finitely generated projective ℂH-module is a finite dimensional complex H-representation. K_{0}(ℂH) can be identified with the complex representation ring and mapping a representation V to its character χ_{V} gives a map

K_{0}(ℂH) → ℂconH.

Now the projection corresponding to an isotypical summand W of the left regular representation is given by

p_{W} = \frac{dim_{ℂ} W}{|H|} ∑_{h∈H} \overline{χ_{W} (h)}h ∈ ℂH,

compare [79]. The χ_{W} are integer multiples of a basis of the free abelian group K_{0}(ℂH). The above formula tells us that up to a base change g ↦ g^{−1} and constants, the class [W] is mapped to χ_{W} under the Hattori-Stallings rank ch_{0}(H). It is well known that the χ_{W} constitute a basis of ℂconH. We see that ch_{0}(H)_{ℂ} is an isomorphism.

Note that the map f in the proof above is in general not injective. In fact there are examples of groups where j_{Γ} is known to be surjective, but K_{0}(ℂΓ) contains torsion [44]. In these cases injectivity of f would imply that j_{Γ} is an isomorphism, but f maps to a vector space and can therefore not be injective. The original isomorphism conjecture [27] deals with the integral group ring ℤΓ. In that case the family of finite subgroups has to be replaced by the larger family of virtually cyclic subgroups. It seems that much more is known about the isomorphism conjecture for ℤΓ because the K-groups of integral group rings are more directly related to topology. In our case the following gives some evidence for the isomorphism conjecture. The first statement is a special case of Moody’s induction theorem.

Theorem 5.20. (i) Let Γ be a polycyclic-by-finite group, then the natural map

j_{Γ} : colim_{H∈FinΓ}K_{0}(ℂH) → K_{0}(ℂΓ)

is surjective and rationally injective.

(ii) The same holds for virtually free groups.

Proof. (i) For a virtually polycyclic group Γ the group ring is noetherian (see 8.12) and of finite global dimension [77, Theorem 8.2.20]. Therefore the category of finitely generated modules coincides with the category of modules which admit a resolution of finite length by finitely generated projective modules. By the resolution theorem [74, 3.1.13] we get that the natural map K_{0}(ℂΓ) → G_{0}(ℂΓ) is an isomorphism. Now the above follows from the general formulation of Moody’s induction theorem, see Theorem 8.22, and the above Proposition 5.19. (ii) See [48, Lemma 4.8.]. Note that a virtually free group is always a finite extension of a free group.

6 Atiyah’s Conjecture for the Free Group on Two Generators

In this section we will prove the Atiyah conjecture in the case where Γ = ℤ∗ℤ is a free group on two generators. The proof uses the notion of Fredholm modules which in this form goes back to Mishchenko [59]. The first example in Connes’s programme for a non-commutative geometry in [17] is a new conceptional proof of the Kadison conjecture for the free group using Fredholm modules. The geometry of the particular Fredholm module used in that case was clarified by Julg and Valette in [39]. The Kadison conjecture says that the reduced C^{∗}-algebra C_{red}^{∗}(Γ) of a torsionfree group has no nontrivial projections. In [48] Linnell observed that the same technique applies to give a proof of Atiyah’s conjecture. We will start with some notions from non-commutative geometry.

6.1 Fredholm Modules

Let H be a Hilbert space and B(H) be the algebra of bounded linear operators on that Hilbert space. We denote by L^{0}(H) and L^{1}(H) the ideal of operators of finite rank respectively the ideal of trace class operators. For p ∈ [1, ∞) we have the Schatten ideals L^{p}(H). The usual trace of a trace class operator a ∈ L^{1}(H) is denoted by tr(a). The ideal of compact operators is denoted by K. Let B be a ∗-algebra. A B-representation on H is by definition a ∗-homomorphism ρ : B → B(H). We explicitly allow that ρ(1) ≠ id_{H}. For our purposes we use a slightly modified notion of Fredholm modules. To stress the difference let us start with a definition which is ideal for theoretical purposes, but unfortunately not very realistic.

Definition 6.1. A (1-summable) Fredholm module (H, ρ_{+}, ρ_{−}) consists of two representations ρ_{±} : B → B(H), such that ρ_{+}(b) − ρ_{−}(b) ∈ L^{1}(H) for all b ∈ B.

For us a Fredholm module for B is basically a tool designed to produce a homomorphism τ : K_{0}(B) → ℝ. Namely one verifies (see 6.3 below) that

τ : K_{0}(B) → ℝ p = (p_{ij}) ↦ ∑_{i} tr(ρ_{+}(p_{ii}) − ρ_{−}(p_{ii}))

defines such a homomorphism. It may be hard or even impossible to construct a nontrivial Fredholm module in this sense for B = C_{red}^{∗}Γ or B = N Γ, compare [18]. To attack the Kadison conjecture one is more modest and restricts oneself to a dense subalgebra B ⊂ C_{red}^{∗}Γ which is closed under holomorphic function calculus, because for such an algebra it is known that K_{0}(B) = K_{0}(C_{red}^{∗}Γ). For purposes of the Atiyah conjecture we do not need B to be in any sense dense in N Γ, but we have to make sure that we can apply τ to all the projections we are interested in, namely those of the form p_{ker a} for a ∈ ℤΓ. The following definition is therefore suitable for our purposes. We will see later that 0-summability is crucial in order to show that we can handle the relevant projections.

Definition 6.2. Let A be a von Neumann algebra and B ⊂ A an arbitrary ∗-closed subalgebra. A p-summable (A, B)-Fredholm module (H, ρ_{+}, ρ_{−}) consists of two A-representations ρ_{±} : A → B(H), such that ρ_{+}(a) − ρ_{−}(a) ∈ L^{p}(H) for all a ∈ B.

In the next subsection we will construct a 0-summable (N Γ, ℂΓ)-Fredholm module. Note that the task of finding a p-summable module becomes more difficult when p becomes smaller. But once we have such a module the following holds.

Proposition 6.3. Let ρ_{±} : A → B(H) be a p-summable (A, B)-Fredholm module.

(i) For q ∈ {0} ∪ [1; ∞) the set A^{q} = {a ∈ A | ρ_{+}(a) − ρ_{−}(a) ∈ L^{q}(H)} is a ∗-subalgebra of A and we have inclusions A^{0} ⊂ A^{1} ⊂ A^{q} ⊂ A^{q′} ⊂ A for q ≤ q′.

(ii) If the Fredholm module is p-summable, then B ⊂ A^{p}.

(iii) The linear map

τ : A^{1} → ℂ a ↦ tr(ρ_{+}(a) − ρ_{−}(a))

has the trace property τ(ab) = τ(ba) for all a, b ∈ A^{1}.

(iv) Tensoring the (A, B)-Fredholm module ρ_{±} with the standard representation ρ_{0} of M_{n}(ℂ) on ℂ^{n} yields an (M_{n}(A), M_{n}(B))-Fredholm module. We have M_{n}(A^{p}) ⊂ M_{n}(A)^{p} and ρ_{±} ⊗ ρ_{0} is again p-summable.

(v) The map τ extends to matrix algebras by

τ : M_{n}(A^{1}) → ℂ (a_{ij}) ↦ ∑_{i} τ(a_{ii})

and yields a homomorphism

τ : K_{0}(A^{1}) → ℝ, p ↦ τ(p).

Proof. Since L^{p}(H) is an ideal the first statement follows from

ρ(ab) − ρ_{−}(ab) = ρ_{+}(a) [ρ_{+}(b) − ρ_{−}(b)] + [ρ_{+}(a) − ρ_{−}(a)] ρ_{−}(b).

If we now apply tr to this equation and use that tr(cd) = tr(dc) for a trace class operator c ∈ L^{1}(H) we see that τ indeed has the trace property. The statement about matrix algebras follows from L^{p}(H)⊗M_{n}(ℂ) ⊂ L^{p}(H ⊗ℂ^{n}) and the definitions.

Since we want to show that certain numbers are integers the following observation about the difference of projections will become a key ingredient.

Lemma 6.4. Let p and q ∈ B(H) be two projections for which p−q ∈ L^{1}(H), then tr(p − q) is an integer.

Proof. The key observation is that there is a selfadjoint compact operator which commutes with p and q at the same time, namely (p − q)^{2}. Let

H = ⨁_{λ≠0} E_{λ} ⊕ ker(p − q)^{2}

be a decomposition of H into the Hilbert sum of eigenspaces of (p − q)^{2}. Since p and q commute with (p − q)^{2} they both respect this decomposition. We will now compute the trace of (p − q) with respect to a basis which is compatible with the above decomposition. Since ker(p − q) = ker(p − q)^{2} only the eigenspaces E_{λ} for eigenvalues λ ≠ 0 contribute. Since they are all finite dimensional we get

tr(p − q) = ∑_{λ≠0} tr((p − q)|_{E_{λ}}) = ∑_{λ≠0} tr(p|_{E_{λ}}) − tr(q|_{E_{λ}}).

On the right hand side we have differences of dimensions of vector spaces and therefore integers. Since the left hand side is finite, only finitely many of these differences can be nonzero.

In our situation we get:

Corollary 6.5. If (H, ρ_{+}, ρ_{−}) is a Fredholm module and if p ∈ A^{1} is a projection, then τ(p) is an integer. The image of τ : K_{0}(A^{1}) → ℝ lies in ℤ.

Proof. ρ_{+}(p) and ρ_{−}(p) are projections, and by definition of A^{1} the difference ρ_{+}(p) − ρ_{−}(p) is a trace class operator. The above lemma applies.

For an operator a ∈ B(H) denote the orthogonal projection onto its kernel by p_{ker a}. The passage from a to p_{ker a} is of particular importance for the Atiyah conjecture. The following lemma explains why we are so much interested in 0-summable Fredholm modules.

Lemma 6.6. Let a, b ∈ B(H) be operators with a − b ∈ L^{0}(H), then also p_{ker a} − p_{ker b} ∈ L^{0}(H).

Proof. Let K = ker(a − b) and let K^{⊥} be the orthogonal complement in H. We have a|_{K} = b|_{K} and ker a∩K = ker b∩K and therefore also p_{ker a}|_{ker a∩K} = p_{ker b}|_{ker a∩K}. Now let K′ be an orthogonal complement of ker a ∩ K in K, then K′ is perpendicular to ker a and ker b, and thus we have p_{ker a}|_{K′} = 0 = p_{ker b}|_{K′}. We see that p_{ker a} and p_{ker b} differ only on the space K^{⊥} which is finite dimensional by assumption. In particular p_{ker a} − p_{ker b} ∈ L^{0}(H).

6.2 Some Geometric Properties of the Free Group

Let Γ = ℤ ∗ ℤ =< s, t | > be a free group on two generators s and t. The Cayley graph with respect to this set of generators is a tree. The group Γ operates on this tree. On the set of vertices V the operation is free and transitive, on the set E of edges the operation is also free but there are as many orbits as there are generators. Therefore there are isomorphisms V ≅ Γ and E ≅ Γ ⊔ Γ of Γ-sets. The tree is turned into a metric space by identifying each edge with a unit interval. As a metric space this tree has the pleasant property, that given any two distinct points x ≠ y there exists a unique geodesic joining the two. We will denote this geodesic symbolically by x → y. Now given a geodesic x → y there is a unique edge Init(x → y) ∈ E, the initial edge, which meets the geodesic and x at the same time. This allows us to define the following map

f : V → E ⊔ {pt}
x ↦ Init(x → x_{0}) if x ≠ x_{0}
x_{0} ↦ pt.

Here x_{0} ∈ V is an arbitrary but fixed vertex. This map is a bijection and it is almost Γ-equivariant in the following sense:

Lemma 6.7. For a fixed g ∈ Γ there is only a finite number of vertices x ≠ x_{0} with gf(x) ≠ f(gx). The number of exceptions equals the distance from x to gx_{0}.

Proof. Since we are dealing with a tree we have gInit(x → x_{0}) = Init(gx → gx_{0}) ≠ Init(gx → x_{0}) if and only if gx ∈ (gx_{0} → x_{0}).

It is shown in [22] that free groups are the only groups which admit maps with properties similar to those of the map f above.

6.3 Construction of a Fredholm Module

We will now use the map f : V → E ⊔ {pt} from the preceding subsection to construct a 0-summable (N Γ, ℂΓ)-Fredholm module for which the trace τ coincides with the standard trace on N Γ we are interested in. Let l^{2}V and l^{2}E be the free Hilbert spaces on the sets V and E. From the preceding subsection we know that we have isomorphisms of ℂΓ-representations l^{2}V ≅ l^{2}Γ and l^{2}E ≅ l^{2}Γ ⊕ l^{2}Γ. Therefore both representations extend to N Γ-representations. Let ℂ denote the very trivial one-dimensional representation with ax = 0 for all a ∈ N Γ and x ∈ ℂ. If we consider the map f above as a bijection of orthonormal bases (where {pt} is the basis for ℂ) we get an isometric isomorphism of Hilbert spaces

F : l^{2}V → l^{2}E ⊕ ℂ.

Let ρ_{+} : N Γ → B(l^{2}V) be the representation on l^{2}V and ρ be the one on l^{2}E ⊕ ℂ. We define ρ_{+} as the pull back of ρ to l^{2}V via F, i.e. ρ_{−}(a) = F^{−1}ρ(a)F.

Proposition 6.8. (i) The representations ρ_{+} and ρ_{−} define a 0-summable (N Γ, ℂΓ)-Fredholm module.

(ii) For all a ∈ N Γ we have

tr_{N Γ}(a) = ∑_{x∈V} < (ρ_{+}(a) − ρ_{−}(a))x, x > .

(iii) For a ∈ M_{n}(N Γ^{1}) we have tr_{N Γ}(a) = τ(a).

Proof. We start with the second statement. Since Γ operates freely on V and E we get for g ∈ Γ

< [ρ_{+}(g) − ρ_{−}(g)] x_{0}, x_{0} > = < ρ_{+}(g)x_{0}, x_{0} > − < ρ(g)f(x_{0}), f(x_{0}) >
= δ_{g,e} = tr_{N Γ}(g)

and for x ∈ V with x ≠ x_{0}

< [ρ_{+}(g) − ρ_{−}(g)] x, x > = < ρ_{+}(g)x, x > − < ρ(g)f(x), f(x) >
= δ_{g,e} − δ_{g,e} = 0.

Summing over x ∈ E we get the claim first for g ∈ Γ and then by linearity for a ∈ ℂΓ. The claim follows if we can show that tr_{N Γ}(a) and < [ρ_{+}(a) − ρ_{−}(a)] x, x > are continuous functionals with respect to the weak (operator) topology on N Γ, because N Γ is the weak closure of ℂΓ. Now tr_{N Γ}(a) =< ae, e > is weakly continuous, since | < . e, e > | is one of the defining seminorms for the weak topology. One easily verifies that ρ_{+} and ρ and therefore also ρ_{−} and ρ_{+} − ρ_{−} are weak-weak continuous. Since | < . x, x > | is again a defining seminorm for the weak topology the claim follows. For the first statement we have to show that for every a ∈ ℂΓ the operator ρ_{+}(a)−ρ_{−}(a) has finite rank. By linearity it is sufficient to treat ρ_{+}(g)−ρ_{−}(g) for g ∈ Γ. By Lemma 6.7 we have

ρ_{+}(g)x = gx = f^{−1}(gf(x)) = (F^{−1} ∘ ρ(g) ∘ F)x = ρ_{−}(g)x

for x ∈ V with finitely many exceptions. The last statement follows immediately from the second.

We need one further lemma.

Lemma 6.9. If a lies in N Γ^{0}, then also the projection p_{ker a} lies in N Γ^{0}.

Proof. By definition of N Γ^{0} we know that ρ_{+}(a) − ρ_{−}(a) is a finite rank operator. From 6.6 we know that also p_{ker ρ_{+}(a)} − p_{ker ρ_{−}(b)} is of finite rank. Since up to the degenerate summand ℂ we are dealing with direct sums of the left regular representation we get

ρ_{+}(p_{ker a}) = p_{ker ρ_{+}(a)} respectively ρ_{−}(p_{ker a}) + p_{ℂ} = p_{ker ρ_{−}(a)}

where p_{ℂ} is the projection onto ℂ. The claim follows.

Theorem 6.10. Let Γ = ℤ ∗ ℤ be the free group on two generators and a ∈ M_{n}(ℂΓ). Let p_{ker a} denote the orthogonal projection onto ker a ⊂ l^{2}Γ^{n}, then tr_{Γ}(p_{ker a}) is an integer.

Proof. By 0-summability we know ℂΓ ⊂ N Γ^{0} and therefore

M_{n}(ℂΓ) ⊂ M_{n}(N Γ^{0}) ⊂ M_{n}(N Γ)^{0}.

Let a be in M_{n}(ℂΓ). From the matrix analogue of the preceding lemma we know that p_{ker a} ∈ M_{n}(N Γ)^{0}. By Proposition 6.8 we know that tr_{N Γ}(p_{ker a}) = τ(p_{ker a}) which is an integer by Corollary 6.5.

Corollary 6.11. The Atiyah conjecture with complex coefficients holds for the free group on two generators.

Proof. We have dim_{U}(ker(a ⊗ id_{UΓ})) = tr_{A}(p_{ker a}) by definition of dim_{U}.

7 Classes of Groups and Induction Principles

The Atiyah conjecture is proven in [48] for groups in a certain class C (Definition 7.5), which in addition have a bound on the orders of finite subgroups. In this section we will review the class C and prove that it is closed under certain processes (Theorem 7.7). The class C contains the class of elementary amenable groups. So we begin by collecting some facts about elementary amenable groups and the related class of amenable groups.

7.1 Elementary Amenable Groups

Definition 7.1. The class EG of elementary amenable groups is the smallest class of groups with the following properties:

(i) EG contains all finite groups and all abelian groups.

(ii) EG is closed under directed unions: If Γ = ⋃_{i∈I} Γ_{i}, where the Γ_{i}, i ∈ I form a directed system of subgroups (given i, j ∈ I there exists k ∈ I, such that Γ_{i} ⊂ Γ_{k} and Γ_{j} ⊂ Γ_{k} ) and each Γ_{i} belongs to EG, then Γ belongs to EG.

(iii) EG is closed under extensions: If 1 → H → Γ → G → 1 is a short exact sequence such that H and G are in EG, then Γ is in EG.

(iv) EG is closed under taking subgroups.

(v) EG is closed under taking factor groups.

It is shown by Chou in [12] that the first three properties suffice to characterize the class EG. That EG is closed under taking subgroups and factor groups follows from the other properties. Chou also gives a description of the class of elementary amenable groups via transfinite induction. Following [43] we will give a similar description below. But first we need some notation. If X and Y are classes of groups, let X Y denote the class of X -by-Y groups, that means the class of those groups Γ, for which there is an exact sequence 1 → H → Γ → G → 1 with H in X and G in Y. Similarly let LX denote the class of locally-X -groups: A group Γ is in LX if every finitely generated subgroup of Γ is in X.

Proposition 7.2. The class of elementary amenable groups has the following description via transfinite induction: Let B denote the class of finitely generated abelian-by-finite groups and let for every ordinal α the class D_{α} be defined inductively as follows:

• D_{0} = 1.

• D_{α} = (LD_{α−1})B if α is a successor ordinal.

• D_{α} = ⋃_{β<α} D_{β} if α is a limit ordinal.

The class of elementary amenable groups EG coincides with ⋃_{α≥0} D_{α}. Moreover every class D_{α} is subgroup closed and closed under extensions by finite groups.

Proof. See Lemma 3.1 in [43].

Whenever one wants to prove a statement about elementary amenable groups one can use such a description to give an inductive proof. It is therefore desirable to have small induction steps. On the other hand it might be useful if the intermediate classes D_{α} have good properties, as for example being subgroup-closed or being closed under extensions by finite groups. In easy cases such an inductive proof comes down to the following.

Proposition 7.3 (Induction Principle for EG). Suppose P is a property of groups such that:

(I) P holds for the trivial group.

(II) If 1 → H → Γ → A → 1 is an exact sequence, where A is finitely generated abelian-by-finite and P holds for H, then it also holds for Γ.

(III) If Γ = ⋃_{i∈I} Γ_{i} is the directed union of the subgroups Γ_{i} and P holds for all the Γ_{i}, then P holds for Γ.

Then P holds for all elementary amenable groups. Instead of (II) it would also be sufficient to verify:

(II)’ If 1 → H → Γ → A → 1 is an exact sequence, where A is finite or cyclic, and P holds for H, then it also holds for Γ.

Proof. This follows easily via transfinite induction from the description of the class given in Proposition 7.2. Since one extension by a finitely generated abelian-by-finite group can always be replaced by finitely many extensions by infinite cyclic groups followed by one extension by a finite group (II)’ implies (II).

Before we go on and introduce Linnell’s class C, we will discuss the class of amenable groups.

Definition 7.4. A group Γ is called amenable, if there is a Γ-invariant linear map μ : l^{∞}(Γ, ℝ) → ℝ, such that

inf{f(γ)|γ ∈ Γ} ≤ μ(f) ≤ sup{f(γ)|γ ∈ Γ} for f ∈ l^{∞}(Γ).

Here l^{∞}(Γ, ℝ) is the space of bounded real valued functions on Γ. The map μ is called an invariant mean. The class of amenable groups is denoted by AG.

The class of amenable groups AG has the five properties that were used to define the class of elementary amenable groups [20], therefore EG ⊂ AG. The simplest example of a group which is not amenable is the free group on two generators [71, p.6]. Let N F denote the class of groups which do not contain a free subgroup on two generators. Since AG is subgroup closed AG ⊂ N F. Day posed the question whether EG = AG or AG = N F. The question remained open for a long time. Meanwhile there are examples of groups (even finitely presented) showing EG ≠ AG due to Grigorchuk [32]. Examples by Olshanski [68] show AG ≠ N F. These examples are finitely generated. Whether there is a finitely presented group which is not amenable, but does not contain a free subgroup on two generators, is an open question. There are many other characterizations of amenable groups. In particular it seems that amenability of a group Γ has a strong impact on the different algebras of operators associated to the group. Here are some facts. For more details and references see [71].

(i) The invariant mean in the definition of amenability can as well be characterized as a Γ-invariant bounded linear functional μ on l^{∞}(Γ, ℂ) equipped with supremum-norm, such that μ(1) = 1 and μ(f) ≥ 0 for every f ≥ 0. In the language of operator algebras such a μ is called a (Γ-invariant) state on the commutative C^{∗}-algebra l^{∞}(Γ, ℂ).

(ii) There is the so called Fölner criterion which characterizes amenability of a group in terms of the growth of its Cayley graph, see [28] and [3, Theorem F.6.8, p.308].

(iii) The natural map C_{max}^{∗}(Γ) → C_{red}^{∗}(Γ) from the maximal C^{∗}-algebra to the reduced C^{∗}-algebra of the group is an isomorphism if and only if the group Γ is amenable, see [72, Theorem 7.3.9, p. 243].

(iv) There is a notion of amenability for von Neumann algebras and C^{∗}- algebras. Group von Neumann algebras and group C^{∗}-algebras of amenable groups are special examples ([71, 1.31 and 2.35]).

(v) If Γ is amenable the group von Neumann algebra N Γ is very close to being flat over the group ring ℂΓ, namely it is shown in [52, Theorem 5.1] that dim_{N Γ} Tor_{ℂΓ}^{p}(N Γ, M) = 0 for all p ≥ 1. Compare also Theorem 8.4.

7.2 Linnell’s Class C

In [48] Linnell introduced the following class of groups:

Definition 7.5. Let C denote the smallest class of groups, with the following properties:

(i) C contains all free groups.

(ii) C is closed under extensions by elementary amenable groups: If there is a short exact sequence 1 → H → Γ → G → 1 with H in C and G in EG, then Γ belongs to C.

(iii) C is closed under directed unions: If Γ = ⋃_{i∈I} Γ_{i}, where the Γ_{i}, i ∈ I form a directed system of subgroups (given i, j ∈ I there exists k ∈ I, such that Γ_{i} ⊂ Γ_{k} and Γ_{j} ⊂ Γ_{k} ) and each Γ_{i} belongs to C, then Γ belongs to C.

The class of groups in C which have a bound on the orders of finite subgroups will be denoted by C′.

Again there is an inductive description of this class of groups. Concerning notation compare the remarks following Definition 7.1.

Proposition 7.6. The class C has the following description via transfinite induction: Let B denote the class of finitely generated abelian-by-finite groups and let for every ordinal α the class C_{α} be defined inductively as follows:

• C_{0} is the class of free-by-finite groups.

• C_{α} = (LC_{α−1})B if α is a successor ordinal.

• C_{α} = ⋃_{β<α} C_{β} if α is a limit ordinal.

The class C coincides with ⋃_{α≥0} C_{α}. Moreover, every class C_{α} is subgroup closed and closed under extensions by finite groups.

Proof. See [48].

Given a concrete group it is often difficult to tell whether or not it belongs to the class C. It is therefore useful to know that C is closed under other processes than the defining ones.

Theorem 7.7. (i) The class C is closed under taking subgroups.

(ii) If Γ is a group in C and A is an elementary amenable normal subgroup, then the quotient Γ/A is in C.

(iii) If 1 → Z → Γ → G → 1 is an exact sequence of groups where G is in C and Z is finite or cyclic, then Γ is in C. More generally, this holds if Z is any elementary amenable group whose automorphism group is also elementary amenable.

(iv) The class C is closed under taking arbitrary free products: If Γ_{i}, i ∈ I is any collection of groups in C, then the free product

Γ := ∗_{i∈I}Γ_{i}

also lies in C. If all the factors have a bound on the orders of finite subgroups (Γ_{i} ∈ C′) and if there are only finitely many factors, then Γ is in C′.

This allows us to give some concrete examples of interesting groups in the class C.

Example 7.8. It is well known that P SL_{2}(ℤ) ≅ ℤ/2ℤ∗ℤ/3ℤ. Theorem 7.7 immediately yields that P SL_{2}(ℤ) lies in the class C and also that SL_{2}(ℤ) and GL_{2}(ℤ) are in C.

The rest of this section will be concerned with the proof of Theorem 7.7. Part (i) follows immediately from the fact that all the C_{α} in Proposition 7.6 are closed under taking subgroups. The proof of (ii), (iii) and (iv) is based on the following induction principle which will be used repeatedly.

Proposition 7.9 (Induction Principle for C). Suppose P is a property of groups such that:

(I) P holds for free groups.

(II) If there is a short exact sequence 1 → H → Γ → A → 1 with A elementary amenable and if the property holds for H, then it is also holds for Γ.

(III) If Γ is the directed union of subgroups Γ_{i} and the property holds for every Γ_{i}, then it holds for Γ.

In this case P holds for every group in the class C. Instead of (II) it is also sufficient to check one of the following statements:

(II)’ If there is a short exact sequence 1 → H → Γ → A → 1 with A finitely generated abelian-by-finite and if the property holds for H, then it also holds for Γ.

(II)” If there is a short exact sequence 1 → H → Γ → A → 1 with A finite or infinite cyclic and if the property holds for H, then then it also holds for Γ.

Proof. This follows in the (II)’-case by transfinite induction from the description of the class C given in Proposition 7.6. Since (II) is stronger than (II)’ it remains to be shown that also (II)” is sufficient. This follows from the fact that one can split an extension by one finitely generated abelian-by-finite group into finitely many extensions by infinite cyclic groups followed by one extension by a finite group.

Proof of 7.7(ii). We use the induction principle. Let P (Γ) be the following statement: For every elementary amenable normal subgroup A ◁ Γ the quotient Γ/A is in C.

(I) Let Γ be a free group. Since any subgroup of a free group is free, an elementary amenable subgroup A must be either trivial or infinite cyclic. Suppose A is infinite cyclic. Since every finitely generated normal subgroup of a free group has finite index ([57, Chapter I, Proposition 3.12.]), the group Γ/A is finite and therefore in C. (II) Let 1 → H → Γ → A′ → 1 be an exact sequence with A′ elementary amenable and suppose P (H) holds. Let A be an elementary amenable normal subgroup of Γ. The following diagram is commutative and has exact rows and columns.

[figure: A ∩ H; ⊂; H; H/A ∩ H]

[figure: A; ∩; ⊂; Γ; ∩; Γ/A; ∩]

[figure: A′; =; A′]

As a subgroup of A, the group A ∩ H is elementary amenable. By P (H) the group H/A ∩ H is in C. Since Γ/A is an extension of H/A ∩ H by the elementary amenable group A′ we see, that Γ/A is in C. (III) Let Γ = ⋃_{i∈I} Γ_{i} be a directed union, suppose P (Γ_{i}) holds for all i ∈ I and let A ◁ Γ be an elementary amenable normal subgroup. The quotient Γ/A is the directed union of the Γ_{i}/A ∩ Γ_{i} which are in C since A ∩ Γ_{i} is elementary amenable as a subgroup of A and P (Γ_{i}) holds by assumption. This completes the proof of 7.7 (ii).

Proof of Theorem 7.7(iii). The proof of is very similar. Let now P (G) be the statement: If 1 → Z → Γ ^{p}^{}G → 1 is an exact sequence with Z and if Aut(Z) is elementary amenable, then Γ is in C. (I) If G is a free group, then the sequence splits. A section determines a homomorphism α : G → Aut(Z) and Γ is a semidirect product Γ ≅ Z ⋊_{α} G.

In the commutative diagram

im(α)
[figure: α ∘ p]
[figure: Z; ⊂; Γ; p; G; α]

[figure: Z; ⊂; ker(α ∘ p); ∪; ker(α); ∪]

the group ker(α ∘ p) is a direct product of Z and the free group ker(α) and is therefore in C. Now Γ is an extension of ker(α ∘ p) by im(α) ⊂ Aut(Z), which is by assumption elementary amenable. We conclude that Γ is in C. (II) Let 1 → H → G → A → 1 be an exact sequence with A elementary amenable and suppose P (H) holds. In the commutative diagram

Z  =  Z

[figure: p^{−1}(H); ∩; ⊂; Γ; ∩; A]

[figure: H; ⊂; G; A]

the group p^{−1}(H) is in C because P (H) holds, and Γ is an extension of this group by an elementary amenable group. Therefore Γ is in C and P (G) holds. (III) Let G be the directed union of the groups G_{i}, i ∈ I. In the exact sequence 1 → Z → p^{−1}(G_{i}) → G_{i} → 1 the middle group is in C by the assumption P (G_{i}). Now Γ is the directed union of the p^{−1}(G_{i}) and is therefore in C. This finishes the proof of 7.7 (iii).

We will now go on to prove the statement (iv) about free products in Theorem 7.7. We need the following observation about extensions of free products.

Lemma 7.10. Let 1 → K → G ⟶^{p} Q → 1 be a short exact sequence of groups and let {g_{q}, q ∈ Q} ⊂ G be a fixed set of representatives for the cosets. Then for any group H the kernel of the homomorphism

G ∗ H ⟶^{p∗1} Q

is the free product

(∗_{q∈Q}H^{g_{q}}) ∗ K.

Here H^{g} is the conjugate of H by g in G ∗ H. In particular, we have an extension

1 → (∗_{q∈Q}H) ∗ K → G ∗ H → Q → 1

Proof. Note that by definition of a free product every element x ∈ G ∗ H can be written uniquely as a word x = g_{1}h_{1}g_{2}h_{2}·. . .·g_{n}h_{n} with letters from G and H, where g_{2}, …, g_{n} ≠ 1 and h_{1}, …, h_{n−1} ≠ 1. Every element g ∈ G can be written uniquely as g = kg_{q} for some k ∈ K and q ∈ Q. Define inductively k_{i} and q_{i} such that g_{1} = k_{1}g_{q_{1}}, g_{q_{1}}g_{2} = k_{2}g_{q_{2}}, …, g_{q_{n−1}}g_{n} = k_{n}g_{q_{n}}. This allows us to rewrite x as follows:

x = k_{1}(g_{q_{1}}h_{1}g_{q_{1}}^{−1})k_{2}(g_{q_{2}}h_{2}g_{q_{2}}^{−1})k_{3} · . . . · k_{n}(g_{q_{n}}h_{n}g_{q_{n}}^{−1})g_{q_{n}}.

Now suppose x is in the kernel of p ∗ 1. Since p ∗ 1(x) = g_{q_{n}} = 1 we see that x can be written as a word with letters in the H^{g_{q}} and in K as proposed by the brackets. Suppose there is another such spelling of x as a word with letters in the H^{g_{q}} and in K

x = (k′_{1}g_{q′_{1}})h′_{1}(g_{q′_{1}}^{−1}k′_{2}g_{q′_{2}})h′_{2}(g_{q′_{2}}^{−1}k′_{3}g_{q′_{3}}) · . . . · h′_{m}(g_{q′_{m}}^{−1}).

Reading this as indicated by the brackets as a word with letters in G and H and using the above mentioned uniqueness yields m = n and g_{1} = k′_{1}g_{q′_{1}}, h′_{1} = h_{1}, g_{2} = g_{q′_{1}}^{−1}k′_{2}g_{q′_{2}}, h′_{2} = h_{2}, …, h′_{n} = h_{n}, g_{q′_{n}}^{−1} = 1. Uniqueness of the spelling as a word with letters in the H^{g_{q}} and K follows inductively from this, making use of the uniqueness of the k_{i} and q_{i}. This proves that the kernel is the free product ∗_{q∈Q}H^{g_{q}} ∗ K.

Lemma 7.11. Let Γ be in C and F be any free group. Then Γ ∗ F is in C.

Proof. We use the induction principle. (I) The statement is clear if Γ is a free group. (II) Now take an extension 1 → H → Γ → A → 1 where A is elementary amenable and H ∗ F′ belongs to C for any free group F′. By Proposition 7.10 we get an extension 1 → ∗_{a∈A}F ∗ H → Γ ∗ F → A → 1. Since F′ := ∗_{a∈A}F is again free, this is an extension of a group of C by an elementary amenable group, therefore Γ ∗ F is in C. (III) Let Γ = ⋃_{i∈I} Γ_{i} be a directed union of subgroups and suppose Γ_{i} ∗ F ∈ C for all i ∈ I. Then Γ ∗ F = ⋃_{i∈I} Γ_{i} ∗ F belongs to C since C is closed under directed unions.

Lemma 7.12. Suppose G is in C, and let I be any index set. Then G_{I} := ∗_{i∈I}G is in C.

Proof. Proposition 7.10 with p = id_{ℤ} shows that there is an extension 1 → ∗_{z∈ℤ}G → G ∗ ℤ → ℤ → 1. By Lemma 7.11 the group G ∗ ℤ belongs to the class C. Since C is subgroup closed ∗_{z∈ℤ}G is in C. Again using that C is subgroup closed we see that the free product of finitely many copies of G belongs to C. Since G_{I} = ∗_{i∈I}G is the directed union of free products of finitely many copies of G we are done.

Lemma 7.13. Fix a group Γ ∈ C. Let I be any index set and G_{I} := ∗_{i∈I}G. If G is in C, then also Γ ∗ G_{I}.

Proof. Again we use the induction principle. (I) If Γ is a free group, the statement follows from Lemma 7.12 and lemma 7.11. (II) Suppose 1 → H → Γ → A → 1 is an exact sequence of groups with A elementary amenable. Then we get an extension 1 → H ∗ G_{I′} → Γ ∗ G_{I} → A → 1 with G_{I′} := ∗_{a∈A}(∗_{i∈I}G) another free product of copies of G. If the statement holds for H, then H ∗ G_{I′} is in C since we did not specify the index set. But then also Γ ∗ G_{I} belongs to C. (III) If Γ is a directed union Γ = ⋃_{j∈J} Γ_{j} and Γ_{j} ∗ G_{I} is in C for all j ∈ J, then Γ ∗ G_{I} belongs to C since it is the directed union of the Γ_{j} ∗ G_{I}.

Now we are prepared to finish the proof of Theorem 7.7.

Proof of Theorem 7.7(iv). Since C is closed under taking subgroups 7.13 implies that a free product with two factors lies in C if the factors do. Ordinary induction shows that any finite free product of groups of C belongs to C. Since an arbitrary product ∗_{i∈I}Γ_{i} is the directed union of the subgroups ∗_{i∈J}Γ_{i} with J ⊂ I finite, the group ∗_{i∈I}Γ_{i} lies in C.

8 Linnell’s Theorem

We will now discuss in detail the following result due to Linnell [48].

Theorem 8.1. If Γ is a group in the class C which has a bound on the orders of finite subgroups, then the Atiyah conjecture with complex coefficients (see 5.3) holds for Γ.

For the proof we will follow the strategy outlined in Section 5.2. So our task is to find intermediate rings of the extension ℂΓ ⊂ UΓ with good properties. By definition the class C is closed under taking extensions by finite and infinite cyclic groups. On the ring theoretical side an extension corresponds to a crossed product, see Appendix IV. One should think of this as a process which makes the ring worse, e.g. it raises the homological dimension. The technique to ameliorate the ring will be localization. In a large part of the following we will therefore investigate what happens if one combines and iterates these processes. Directed unions of groups correspond to directed unions of rings and are much easier to handle. The candidates for the intermediate rings are the division closure and the rational closure of ℂΓ in UΓ. The reader who is not familiar with the concepts of division and rational closure should first read the first three subsections of Appendix III. In particular the situation of Proposition 13.17 will occur again and again. We denote by DΓ = D(ℂΓ ⊂ UΓ) the division closure of ℂΓ in UΓ and by RΓ = R(ℂΓ ⊂ UΓ) the rational closure of ℂΓ in UΓ. Since a von Neumann regular ring is division closed and rationally closed (see 13.15) these rings constitute a minimal possible choice for the ring SΓ we are looking for.

Note 8.2. If the intermediate ring SΓ is von Neumann regular, then we have DΓ ⊂ RΓ ⊂ SΓ.

We should mention that in all known cases DΓ and RΓ coincide. We denote the set of elements of ℂΓ which become invertible in UΓ by T(Γ) = T(ℂΓ ⊂ UΓ) = ℂΓ ∩ UΓ^{×}. Similarly the set of those matrices over ℂΓ which become invertible over UΓ is denoted by Σ(Γ) = Σ(ℂΓ ⊂ UΓ). The set of nonzerodivisors of ℂΓ is denoted NZD(ℂΓ). Here is now a precise statement of the result that will be proven. Combined with Theorem 5.10 this implies the above Theorem.

Theorem 8.3. Let Γ be a group in the class C which has a bound on the orders of finite subgroups, then the following holds.

(A) The ring DΓ is semisimple, and the inclusion ℂΓ ⊂ RΓ is universal Σ(Γ)-inverting.

The semisimplicity of DΓ implies DΓ = RΓ, compare 13.16.

(B) The natural map

colim_{K∈Fin(Γ)}K_{0}(ℂK) → K_{0}(DΓ)

is surjective.

With little extra effort we prove the following statement which appears already in [46].

Theorem 8.4. If moreover Γ is elementary amenable and has a bound on the orders of finite subgroups, the following holds.

(C) The set of non-zerodivisors NZD(ℂΓ) equals T(Γ), i.e. every nonzerodivisor of ℂΓ becomes invertible in UΓ. The pair (ℂΓ, T(Γ)) satisfies the Ore condition. The Ore localization is semisimple and isomorphic to the division closure:

DΓ ≅ (ℂΓ)T(Γ)^{−1}.

In particular, in this case ℂΓ ⊂ DΓ is not only universal Σ(Γ)-inverting, but also universal T(Γ)-inverting. An Ore localization with respect to the set of all non-zerodivisors is called a classical ring of fractions.

Of course these theorems are proven via transfinite induction. Compare 7.6 and 7.9. Before we give the details of the proof we will discuss the main ingredients and difficulties. The table below might help the reader to follow the proof. The statements proven below are labelled according to this table. Recall that, given classes of groups X and Y, we denote by X Y the class of X -by-Y groups and by LX the class of locally-X -groups. The subclass X′ ⊂ X consists of those groups which have a bound on the orders of finite subgroups. The class of finite groups is denoted by {finite}. Similarly for other classes of groups. The statement of (A) above for the group Γ is abbreviated to (A)_{Γ}. Similarly (A)_{X}, if (A) holds for all groups in the class X. With this shorthand notation we can summarize the statements proven in the following subsection as follows:

AIa)  (A)_{{free}}
BIb)  (B)_{{free}{finite}}

AIIa) (A)_{Y} ⇒ (A)_{Y{infinite cyclic}}

AIIb) (A)_{Y} ⇒ (A)_{Y{finite}}

BIIa)+b) (Y{finite} = Y, (A)_{Y} and (B)_{Y}) ⇒ (B)_{Y{f.g. abelian-by-finite}}

AIII) (A)_{Y′} and (B)_{Y′} ⇒ (A)_{(LY)′}

BIII) (B)_{Y} ⇒ (B)_{LY} .

We will verify below (page 66) that Theorem 8.3 follows via transfinite induction. Note that the strong hypothesis in BIIa) + b) forces us to start the induction with a class of groups which is closed under finite extensions. Therefore it is not enough to prove (A) and (B) for free groups to start the induction.

(A)  (B)

I. Starting the induction. a) Γ free on two generators.

b) Γ free by finite.

II. Induction step Y → YB: Extensions by H. a) H infinite cyclic.

b) H finite.

a) + b) H finitely generated abelian-by-finite.

III. Induction step Y → LY: Directed unions.

One might wonder whether one could separate the (A)-part from the (B)- part, but the proof of AIII) uses the (B)-part of the induction hypothesis. The (B)-part depends of course heavily on the (A)-part.

That ℂΓ ⊂ RΓ is universal Σ(Γ)-inverting is only important for the proof of BIb). If one is only interested in the Atiyah conjecture one could therefore proceed with the induction steps without this extra statement. We have included it in the induction since it is interesting in its own right and leads for example to exact sequences in K-theory, compare Note 10.9. We have tried to give versions of the statements which use as little hypothesis as possible. The following remarks summarize where deeper results enter the proof.

AIa) The major part of the work has already been done in Section 6 and relies on the Fredholm module techniques described there. It is easy to see that DΓ is a skew field, but to show that ℂΓ ⊂ DΓ is universal Σ(Γ)-inverting requires further results about the group ring of the free group.

BIa) This is trivial once we know DΓ is a skew field since K_{0}(DΓ) = ℤ.

AIb) Extensions by finite groups cause no problems for the (A)-part.

BIb) Here it is required that ℂΓ ⊂ DΓ is universal Σ(Γ)-inverting to apply results of Schofield [78] about the K-theory of universal localizations.

AIIa) This is again easy.

AIIb) This part uses Goldie’s theorem and the structure theory of semisimple rings. To verify the hypothesis of Goldie’s theorem we have to go back to operators and functional analysis and use a result of Linnell [47].

AIIa)+b) Combine the two previous parts.

BIIa)+b) Here it does not seem to be possible to prove BIIa) and BIIb) separately. Moody’s induction theorem is the main ingredient in this part. Moreover this, part forces us to start the induction with a class of groups that is closed under extensions by finite groups.

AIII) This part is straightforward, but the hypothesis about the bound on the orders of finite subgroups enters the proof.

BIII) This is easy.

Proof of Theorem 8.3. We use the description of the class C given in Proposition 7.6: Let Γ be in C′. Choose a least ordinal α such that Γ ∈ C_{α}, and suppose (A) and (B) hold for all C′_{β} with β < α. There are three cases: If α = 0, then C_{α} = C_{0} = {free-by-finite} and the result follows from AIa), AIIb) and BIb). If α is a limit ordinal there is nothing to prove since C_{α} = ⋃_{β<α} C_{β}. If α = β + 1 for some ordinal β we have C_{α} = (LC_{β}){f.g. abelian-by-finite} and by assumption (A)_{C′_{β}} and (B)_{C′_{β}} hold. Now AIII) and BIII) yield (A)_{(LC_{β})′} and (B)_{(LC_{β})′}. Here we used that (LY)′ ⊂ L(Y′) for arbitrary classes Y. Now, since one extension by a finitely generated abelian-by-finite group can always be replaced by finitely many extensions by infinite cyclic groups followed by one extension by a finite group we get from AIIa), AIIb) and (Y{f.g. abelian-by-finite})′ = Y′{f.g. abelian-by-finite} that (A)_{C′_{α}} holds. From Proposition 7.6 we know that C_{β}{finite} = C_{β}. For an arbitrary class Y{finite} = Y implies (LY){finite} = LY and we always have Y′{finite} = (Y{finite})′. Therefore BIIa)+b) applies and yields (B)_{C′_{α}}.

We will see that it is relatively easy to prove the following refinements.

CIIa) (C)_{Y} ⇒ (C)_{Y{infinite cyclic}}

CIIb) (C)_{Y} ⇒ (C)_{Y{finite}}

CIII) (C)_{Y′} and (B)_{Y′} ⇒ (C)_{(LY)′}.

Theorem 8.4 follows using the induction principle for elementary amenable groups 7.3. Note that (C) is false for the free group on two generators since this would imply that DΓ is flat over ℂΓ and therefore (DΓ is semisimple) UΓ is flat over ℂΓ. But this cannot be true since we know that H_{1}(Γ; UΓ) does not vanish.

8.1 Induction Step: Extensions - The (A)-Part

In this section we consider an exact sequence of groups

1 → G → Γ → H → 1.

The ring DΓ will be built up in two steps out of DG. We first consider a crossed product DG ∗ H, and we will show that under certain circumstances DΓ is an Ore localization of DG ∗ H.

First we collect what we have to know about the behaviour of localizations with respect to crossed products. For the definition of a crossed product and our conventions concerning notation we refer to Appendix IV. A ring extension R ∗ G ⊂ S ∗ G is called compatible with the crossed product structures if the inclusion map is a crossed product homomorphism. We emphasize again that Proposition 13.17 is the starting point for most of the considerations below.

Proposition 8.5 (Crossed Products and Localization). Let R ∗ G ⊂ S ∗G be a compatible ring extension of crossed products. The crossed product structure map on both rings will be denoted by μ : G → R ∗ G ⊂ S ∗ G

(i) The intermediate ring generated by D(R ⊂ S) and μ(G) carries a crossed product structure such that both inclusions

R ∗ G ⊂ D(R ⊂ S) ∗ G ⊂ S ∗ G

are compatible with the crossed product structures.

(ii) Similarly the ring generated by R(R ⊂ S) and μ(G) in S ∗ G is a compatible crossed product.

(iii) Let Σ be a set of matrices over (or elements of) R which is invariant under the conjugation maps c_{μ(g)}. Let i : R → R_{Σ} be universal Σ-inverting, then R_{Σ} ∗ G exists together with a crossed product homomorphism

i ∗ G : R ∗ G → R_{Σ} ∗ G.

This map is universal Σ-inverting.

(iv) If R ⊂ D(R ⊂ S) is universal T(R ⊂ S)-inverting, then

R ∗ G ⊂ D(R ⊂ S) ∗ G

is universal T(R ⊂ S)-inverting.

(v) If R ⊂ R(R ⊂ S) is universal Σ(R ⊂ S)-inverting, then

R ∗ G ⊂ R(R ⊂ S) ∗ G

is universal Σ(R ⊂ S)-inverting.

(vi) Let T ⊂ R be a multiplicatively closed subset which is invariant under the conjugation maps c_{μ(g)}. Suppose (R, T) fulfills the right Ore condition, then (R ∗ G, T) fulfills the right Ore condition and

RT^{−1} ∗ G ≅ (R ∗ G)T^{−1}.

Proof. (i) Let μ : G → S ∗ G denote the crossed product structure of S ∗ G. By assumption it differs from the one of R ∗ G only by the inclusion R ∗ G ⊂ S ∗ G. We have to show that the conjugation maps c_{μ(g)} can be restricted to D = D(R ⊂ S) and that the ring generated by D and μ(G) is a free D-module with basis μ(G). For τ there is nothing to be done. By definition D = D(R ⊂ S) is division closed in S, i.e. D ∩ S^{×} ⊂ D^{×}. Let α : S → S be an automorphism with α(R) = R. One verifies that α(D) is division closed in S and contains R. Since D is the smallest ring with these properties we have D ⊂ α(D), and arguing with α^{−1} we get equality. In particular this applies to c_{μ(g)}. Using this one can write every element in the ring generated by D and μ(G) in the form ∑_{g∈G} d_{g}μ(g) with d_{g} ∈ D. Since S ∗ G is a free S-module with basis μ(G) this representation is unique. (ii) The proof is more or less identical with the proof above starting this time with M(R(R ⊂ S)) ∩ GL(S) ⊂ GL(R(R ⊂ S)). (iii) Let i : R → R_{Σ} be universal Σ-inverting. By a assumption c_{μ(g)}(Σ) ⊂ Σ and therefore the composition i ∘ c_{μ(g)} : R → R → R_{Σ} is Σ-inverting. The universal property gives us a unique extension \overline{c_{μ(g)}} : R_{Σ} → R_{Σ} of c_{μ(g)}. We define a multiplication on the free R_{Σ}-module with basis the symbols \overline{μ}(g) with g ∈ G by

r \overline{μ}(g) · r′ \overline{μ}(g′) = r \overline{c_{μ(g)}}(r′) \overline{τ}(g, g′) \overline{μ}(gg′)  (1)

for r, r′ ∈ R_{Σ}. Here τ(g, g′) = μ(g) μ(g′) μ(gg′)^{−1} ∈ R^{×} and \overline{τ} = i ∘ τ. For associativity we have to check (compare 14.2)

\overline{τ}(g, g′) · \overline{τ}(gg′, g′′) = \overline{c_{μ(g)}}(\overline{τ}(g′, g′′)) · \overline{τ}(g, g′g′′)  (2)
c_{\overline{τ}(g,g′)} ∘ \overline{c_{μ(gg′)}} = \overline{c_{μ(g)}} ∘ \overline{c_{μ(g′)}}  (3)

The first equation follows from associativity in R ∗ G and the fact that \overline{c_{μ(g)}} extends c_{μ(g)}. For the second equation note that c_{τ(g,g′)} = c_{μ(g)} ∘c_{μ(g′)} ∘c_{μ(gg′)}^{−1} : R → R also leaves Σ invariant and has by the universal property an extension \overline{c_{τ(g,g′)}} : R_{Σ} → R_{Σ} which by uniqueness coincides with c_{\overline{τ}(g,g′)} : R_{Σ} → R_{Σ}. Arguing with the universal property we verify the second equation. Now as R_{Σ} ∗ G exists as a ring it makes sense to consider c_{\overline{μ}(g)} : R_{Σ} ∗ G → R_{Σ} ∗ G. One checks that c_{\overline{μ}(g)} restricts to R_{Σ} and c_{\overline{μ}(g)} = \overline{c_{μ(g)}}. Similarly the notation \overline{τ} is unambiguous, i.e. (i ∘ τ)(g, g′) = \overline{τ}(g, g′) = \overline{μ}(g) \overline{μ}(g′) \overline{μ}(gg′)^{−1}. The map i ∗ G : R ∗ G → R_{Σ} ∗ G given by ∑ r_{g}μ(g) ↦ ∑ i(r_{g})\overline{μ}(g) is a Σ-inverting ring homomorphism and compatible with the crossed product structures. Remains to be verified that it is also universal Σ-inverting. Let f : R ∗ G → S be a given Σ-inverting homomorphism. Denote by j : R → R ∗ G and j_{Σ} : R_{Σ} → R_{Σ} ∗ G the natural inclusion. The universal property of i : R → R_{Σ} applied to the composition f ∘ j : R → R ∗ G → S gives a homomorphism ϕ_{0} : R_{Σ} → S with ϕ_{0} ∘ i = f ∘ j. Now a ring homomorphism ϕ : R_{Σ} ∗ G → S with ϕ ∘ (i ∗ G) = f and ϕ ∘ j_{Σ} = ϕ_{0} necessarily has to be defined by ϕ(j_{Σ}(r)\overline{μ}(g)) = ϕ_{0}(r)f(μ(g)). One ensures that this indeed defines a ring homomorphism. (iv) Denote by i : R → R_{T} the universal T inverting homomorphism and by j : R → D(R ⊂ S) the inclusion. By assumption the homomorphism ϕ : R_{T} → D(R ⊂ S) given by the universal property is an isomorphism. By (iii) i ∗ G exists and is universal T -inverting. From (i) we know that j ∗ G exists and is T -inverting. The universal property applied to this yields ϕ ∗ G which is an isomorphism.

(v) The same argument as in (iii). (vi) The assumption says that given (a_{g}, s) ∈ (R, T) there exist (b_{g}, t_{g}) ∈ (R, T) such that a_{g}t_{g} = sb_{g}, which becomes s^{−1}a_{g} = b_{g}t_{g}^{−1} in RT^{−1}. Moreover finitely many fractions, e.g. (1, c_{μ(g)^{−1}}(t_{g})) ∈ (R, T) can be brought to a common denominator. So there exist (d_{g}, t) ∈ (R, T) such that c_{μ(g)^{−1}}(t_{g})^{−1} = d_{g}t^{−1}. Using this one turns

s^{−1} [∑ a_{g}μ(g)] = ∑ b_{g}t_{g}^{−1}μ(g)
= ∑ b_{g}μ(g)c_{μ(g)^{−1}}(t_{g})^{−1} = [∑ b_{g}μ(g)d_{g}] t^{−1}

into a proof by avoiding denominators. The Ore localization (R ∗ G)T^{−1} is universal T -inverting. From (iii) we know that R_{T} ∗ G ≅ RT^{−1} ∗ G is also universal T -inverting. Therefore the two rings are isomorphic.

In particular, we obtain in our situation:

Proposition 8.6 (Extensions and Localization). Let 1 → G → Γ → H → 1 be an exact sequence of groups and let μ be a set theoretical section μ : H → Γ.

(i) The ring DG ∗ H generated by DG and μ(H) in UΓ has a compatible crossed product structure. The ring generated by RG and μ(H) has a compatible crossed product structure.

(ii) The crossed product ℂG_{T(G)} ∗ H exists together with a map

ℂG ∗ H → ℂG_{T(G)} ∗ H.

This map is universal T(G)-inverting. Similarly the map

ℂG ∗ H → ℂG_{Σ(G)} ∗ H

exists, is compatible and universal Σ(G)-inverting.

(iii) If ℂG ⊂ DG is universal T(G)-inverting, then

ℂG ∗ H ⊂ DG ∗ H

is universal T(G)-inverting. If ℂG ⊂ RG is universal Σ(G)-inverting, then

ℂG ∗ H ⊂ RG ∗ H

is universal Σ(G)-inverting.

(iv) If (ℂG, T (G)) fulfills the Ore condition and D(G) ≅ (ℂG)T (G)^{−1}, then the pair (ℂG ∗ H, T(G)) fulfills the Ore condition, ℂG ∗ H ⊂ DG ∗ H is universal T(G)-inverting and

DG ∗ H ≅ (ℂG ∗ H)_{T(G)} ≅ (ℂG ∗ H)T(G)^{−1}

is an Ore localization.

Proof. Apply the previous proposition.

The next step is the passage from DG ∗ H to DΓ. We will show that under certain circumstances this is an Ore localization. Finally we are interested in combining these two steps. It is therefore important to gather information on iterated localizations. Note that an analogue of (ii) in the following proposition for elements instead of matrices does not hold.

Proposition 8.7 (Iterated Localization). Let R ⊂ S be a ring extension.

(i) The division closure and rational closure are in fact closure operations, namely

D(D(R ⊂ S) ⊂ S) = D(R ⊂ S),
and R(R(R ⊂ S) ⊂ S) = D(R ⊂ S).

(ii) Let Σ ⊂ M(R) be a set of matrices with Σ ⊂ Σ(R ⊂ S). Let R → R_{Σ} be universal Σ-inverting and R_{Σ} → (R_{Σ})_{Σ(R_{Σ}→S)} be universal Σ(R_{Σ} → S)- inverting, then the composition of these maps is universal Σ(R ⊂ S)- inverting. A little less precise

(R_{Σ})_{Σ(R_{Σ}→S)} ≅ R_{Σ(R⊂S)}.

(iii) Let T ⊂ T(R ⊂ S) be a multiplicatively closed subset. Suppose the pair (R, T) and the pair (RT^{−1}, T(RT^{−1} ⊂ S)) both satisfy the right Ore condition, then also (R, T(R ⊂ S)) satisfies the right Ore condition and RT(R ⊂ S)^{−1} ≅ (RT^{−1})(T(RT^{−1} ⊂ S))^{−1}. Both rings can be embedded into S and hence

R ⊂ (RT^{−1})(T(RT^{−1} ⊂ S))^{−1} = RT(R ⊂ S)^{−1} ⊂ S.

(iv) If moreover T(RT^{−1} ⊂ S) = NZD(RT^{−1}), then T(R ⊂ S) = NZD(R) and RT(R ⊂ S)^{−1} is a classical ring of right fractions.

Proof. (i) If R is division closed in S, then D(R ⊂ S) = R. Since D(R ⊂ S) is by definition division closed in S the claim follows. Similarly for the rational closure. (iii) We start with (a, s) ∈ R × T(R ⊂ S) and have to show that a right fraction s^{−1}a can be written as a left fraction. Since R × T(R ⊂ S) ⊂ RT^{−1} × T(RT^{−1} ⊂ S) and the assumption yields elements (b, t) ∈ R × T and (c, u) ∈ R × T with cu^{−1} ∈ T(RT^{−1} ⊂ S) such that (symbolically)

s^{−1}a = (bt^{−1})(cu^{−1})^{−1} = b(cu^{−1}t)^{−1}.

Note that cu^{−1} ∈ T(RT^{−1} ⊂ S) implies c ∈ T(R ⊂ S) and therefore (b, cu^{−1}t) ∈ R × T(R ⊂ S). This is turned into a proof by avoiding inverses which not yet exist. Once RT(R ⊂ S)^{−1} exists we know that R → RT(R ⊂ S)^{−1} is universal T(R ⊂ S)-inverting. Similarly for R → RT^{−1} and RT^{−1} → (RT^{−1})T(RT^{−1} ⊂ S)^{−1}. Now apply (ii). (iv) It is sufficient to check NZD(R) ⊂ T(R ⊂ S). Let r ∈ NZD(R) be a non-zerodivisor in R. Using the Ore condition for RT^{−1} one verifies that r ∈ NZD(RT^{−1}) = T(RT^{−1} ⊂ S). So r becomes invertible in S.

For the proof of 8.7 (ii) we need some preparations. Two matrices (or elements) a, b ∈ M(R) are called stably associated over R if there exist matrices c, d ∈ GL(R), such that

c ( \begin{matrix}a & 0 \\ 0 & 1_{n} \end{matrix} ) d^{−1} = ( \begin{matrix}b & 0 \\ 0 & 1_{m} \end{matrix} )

with suitable m and n. Note that this is an equivalence relation and this relation respects the property of being invertible. Given a set of matrices Σ ⊂ M(R) we denote by Σ̃ the set of all matrices which are stably associated to a matrix in Σ. The first statement of the following lemma is also known as Cramer’s rule.

Lemma 8.8. (i) Let Σ ⊂ M(R) be a set of matrices and f : R → R_{Σ} be universal Σ-inverting. Every matrix a ∈ M(R_{Σ}) is stably associated over R_{Σ} to a matrix f(b) with b ∈ M(R).

(ii) Let \overline{Σ} = f^{−1}(GL(R_{Σ})) be the so called saturation of Σ, then f : R → R_{Σ} is universal \overline{Σ}-inverting and similarly with Σ̃ instead of \overline{Σ}. For purposes of the universal localization we can always replace Σ by Σ̃ or \overline{Σ}.

(iii) Let Σ, Σ′ ⊂ M(R). Let f : R → R_{Σ} be universal Σ-inverting and let R_{Σ} → (R_{Σ})_{f(Σ′)} be universal f(Σ′)-inverting, then the composition g ∘ f : R → (R_{Σ})_{f(Σ′)} is universal Σ ∪ Σ′-inverting and therefore

(R_{Σ})_{f(Σ′)} ≅ R_{Σ∪Σ′}.

Proof. (i) This can be deduced from a generator and relation construction of R_{Σ}. Compare [78, page 53]. (ii) By definition of \overline{Σ} the map f : R → R_{Σ} is \overline{Σ}-inverting. One verifies that it also has the corresponding universal property. Similarly for Σ̃. (iii) Suppose h : R → S is Σ ∪ Σ′-inverting. The universal property of f : R → R_{Σ} gives a unique map ϕ : R_{Σ} → S which is f(Σ′)-inverting. The universal property of g : R_{Σ} → (R_{Σ})_{f(Σ′)} yields the desired unique Φ : (R_{Σ})_{f(Σ′)} → S.

Now we are prepared to complete the proof of Proposition 8.7.

Proof of Proposition 8.7(ii). Let a be a matrix in Σ(R_{Σ} → S). From 8.8 we know that a (as every matrix over R_{Σ}) is stably associated to a matrix f(b) with b ∈ R. For purposes of the universal localization we can therefore replace Σ(R_{Σ} → S) by f(Σ′) for a suitable set of matrices Σ′ ⊂ R. Compare 8.8. Now by 8.8 we get

(R_{Σ})_{Σ(R_{Σ}→S)} ≅ (R_{Σ})_{f(Σ′)} ≅ R_{Σ∪Σ′}.

One verifies that Σ ∪ Σ′ ⊂ Σ(R → S), i.e R → R_{Σ(R→S)} is Σ ∪ Σ′-inverting. An application of 8.9 yields R_{Σ∪Σ′} ≅ R_{Σ(R→S)}.

The next lemma tells us that in some cases the different localizations coincide.

Lemma 8.9. Let R ⊂ S be a ring extension.

(i) Let Σ ⊂ Σ′ and R → R_{Σ} be universal Σ-inverting. If R → R_{Σ} is Σ′-inverting, then it is universal Σ′-inverting.

(ii) Suppose R ⊂ D(R ⊂ S) is universal T(R ⊂ S)-inverting and von Neumann regular, then D(R ⊂ S) = R(R ⊂ S) and R ⊂ R(R ⊂ S) is universal Σ(R ⊂ S)-inverting.

Proof. (i) Note that f has the universal property. (ii) A von Neumann regular ring is division closed and rationally closed in every overring, compare 13.15. Therefore D(R ⊂ S) = R(R ⊂ S). Since R → R(R ⊂ S) = D(R ⊂ S) is Σ(R ⊂ S)-inverting an application of (i) with Σ = T (R ⊂ S) and Σ′ = Σ(R ⊂ S) yields the result.

Now we apply the above to our situation. The second statement of the following proposition is the abstract set-up for the most difficult part of the induction. The last statement will be applied later only in the case where Γ is elementary amenable.

Proposition 8.10. Let 1 → G → Γ → H → 1 be an exact sequence of groups.

(i) We have

D(DG ∗ H ⊂ UΓ) = DΓ,
and R(RG ∗ H ⊂ UΓ) = RΓ.

(ii) Suppose ℂG → RG is universal Σ(G)-inverting and the pair

(RG ∗ H, T(RG ∗ H ⊂ UΓ))

fulfills the Ore condition. Then

D(RG ∗ H ⊂ UΓ) ≅ (RG ∗ H)T(RG ∗ H ⊂ UΓ)^{−1}.

If this ring is von Neumann regular it coincides with RΓ, and in this case ℂΓ ⊂ RΓ is universal Σ(Γ)-inverting.

(iii) If ℂΓ ⊂ DΓ is universal T(Γ)-inverting and DΓ is von Neumann regular, then DΓ = RΓ and ℂΓ ⊂ DΓ is also universal Σ(Γ)-inverting.

(iv) Suppose (ℂG, T(G)) satisfies the Ore condition and one can show that (DG ∗ H, T(DG ∗ H ⊂ UΓ)) satisfies the Ore condition, then

DΓ ≅ (ℂΓ)T(Γ)^{−1}

is an Ore localization and universal T(Γ)-inverting. If moreover T(DG∗ H ⊂ UΓ) = NZD(DG ∗ H) then T(Γ) = NZD(ℂΓ).

Proof. (i) Since UΓ is von Neumann regular it is division closed, and from the definition of the division closure we get DG = D(ℂG ⊂ UG) = D(ℂG ⊂ UΓ) ⊂ D(ℂΓ ⊂ UΓ) = DΓ. Since D ∗ G is the subring generated by DG and H we have D ∗ G ⊂ DΓ. Now the result follows from 8.7 (i). (ii) From 8.6 (iii) and the assumption we know that f : ℂG ∗ H ⊂ RG ∗ H is universal Σ(G)-inverting. As always the Ore localization RG ∗ H → (RG ∗ H)T(RG ∗ H ⊂ UΓ)^{−1} is universal T(RG ∗ H ⊂ UΓ)-inverting and the map to D(RG ∗ H ⊂ UΓ) given by the universal property is an isomorphism, compare 13.17. If this ring is von Neumann regular we can apply 8.9 (ii) and see that

g : RG ∗ H → R(RG ∗ H ⊂ UΓ) = D(RG ∗ H ⊂ UΓ)

is universal Σ(RG ∗ H ⊂ UΓ)-inverting. Now g ∘ f is an iterated localization as in 8.7 (ii) and the result follows. (iii) This is 8.9 (ii). (iv) Apply 8.7 (iii) with R = ℂG ∗ H, S = UΓ and T = T(G). The second statement is 8.7 (iv).

Remember that we want to prove not only von Neumann regularity but semisimplicity for the rings DΓ. The following criterion tells us that it is sufficient to check the ascending chain condition. Note that taking division and rational closures in our situation always leads to ∗-closed subrings of UΓ.

Lemma 8.11 (∗-Closed Subrings of UΓ). Let Γ be any group.

(i) Every ∗-closed subring of UΓ is semiprime.

(ii) A ring which is semiprime and artinian is semisimple.

Proof. (i) Let a be an operator in UΓ. Since a is densely defined and closed, the closure of the graph of a|_{dom(a^{∗}a)} is the graph of a. Now for x ∈ dom(a^{∗}a) ⊂ dom(a) we have < a^{∗}ax, x >= |ax|^{2} and therefore a^{∗}a = 0 implies a = 0. Now given an ideal I ≠ 0 we have to show that I^{2} ≠ 0. Let a ∈ I be nontrivial, then 0 ≠ a^{∗}a ∈ I and 0 ≠ (a^{∗}a)^{∗}(a^{∗}a) = (a^{∗}a)^{2} ∈ I^{2}. (ii) See Theorem 2.3.10 on page 169 in [76].

Lemma 8.12 (Crossed Products and Chain Conditions). Let R ∗ G be a crossed product.

(i) If R is artinian and if G is finite, then R ∗ G is artinian.

(ii) If R is noetherian and if G is finite,infinite cyclic or more generally polycyclic-by-finite, then R ∗ G is noetherian. If R is noetherian and if G ≅ ℤ is infinite cyclic, then R ∗ G is noetherian.

(iii) If R is semisimple of characteristic 0 and if G is arbitrary, then R ∗ G is semiprime.

(iv) If R is semisimple of characteristic 0 and G finite, then R ∗ G is semisimple.

Proof. If G is finite, a chain of ideals in R ∗ G can be considered as a chain of R-modules in the finitely generated R-module R ∗ G which is artinian respectively noetherian as an R-module. This proves (i) and (ii) if G is finite. (ii) Let G = ℤ be infinite cyclic. Let μ : ℤ → R ∗ G be the crossed product structure map, compare 14.1. Set μ̃(n) = (μ(1))^{n}. This defines a new crossed product structure on R∗ℤ which has the advantage that μ̃ and therefore n ↦ c_{μ̃(n)} is a group homomorphism and the corresponding twisting τ̃ is trivial. With this new crossed product structure R ∗ ℤ is a skew-Laurent polynomial ring. Now we first treat the corresponding skew-polynomial ring with the non-commutative analogue of the Hilbert basis theorem ([76, Prop.3.5.2 on page 395]). The skew-Laurent-polynomial ring is then an Ore localization of the skew polynomial ring and therefore again noetherian by [76, Prop. 3.1.13. on page 354]. A polycyclic-by-finite group (by definition, see [77][page 310]) can be obtained by iterated extensions by infinite cyclic groups followed by an extension by a finite group. Therefore an iterated use of the above arguments finishes the proof. (iii) This is a particular case of Theorem I in [69] since a semisimple ring contains no nilpotent ideals. (iv) Combine (i), (ii) and 8.11(ii).

Now we are ready to prove

AIIa) (A)_{Y} ⇒ (A)_{Y{finite}}

CIIa) (C)_{Y} ⇒ (C)_{Y{finite}}.

Or in words:

Proposition 8.13 ( AIIa) and CIIa) ). Let

1 → G → Γ → H → 1

be an exact sequence of groups with H finite. If DG is semisimple and universal Σ(G)-inverting, then

DΓ = DG ∗ H

is semisimple and universal Σ(Γ)-inverting. If moreover T(G) = NZD(G) and if the pair (ℂG, T(G)) satisfies the Ore condition, then T(Γ) = NZD(Γ), the pair (ℂΓ, T(Γ)) satisfies the Ore condition and DΓ is a classical ring of fractions for ℂΓ.

Proof. From the last lemma we know that DG ∗ H is semisimple and in particular von Neumann regular and division closed (compare 13.15), hence DG ∗ H = D(DG ∗ H ⊂ UΓ) = DΓ by 8.10 (i) and moreover T(DG ∗ H ⊂ UΓ) = (DG ∗ H)^{×}. Again, by semisimplicity DG = RG and DΓ = RΓ, and an application of 8.10 (ii) with T(DG∗H ⊂ UΓ) = (DG∗H)^{×} (nothing needs to be inverted) yields the result. Similarly the second statement follows from 8.10 (iv).

We now turn to the case where H is infinite cyclic. The crossed products which occur in this case are particularly simple, they are so-called skew-Laurent polynomial rings. From Lemma 8.12 about crossed products and chain conditions we know that DG ∗ ℤ is noetherian, ∗-closed and therefore semiprime 8.11(i). The following theorem is therefore very promising.

Theorem 8.14 (Goldie’s Theorem). Let R be a right noetherian semiprime ring. Let T = NZD(R) be the set of all non-zerodivisors in R, then (R, T) satisfies the right Ore condition and the ring RT^{−1} is semisimple.

Proof. See section 9.4 in [16].

The crux of the matter is that we do not know that all non-zerodivisors become invertible in UΓ. A priori we only have T(DG∗ℤ ⊂ UΓ) ⊂ NZD(DG∗ ℤ). Therefore even though we know that the Ore localization

DG ∗ ℤ → (DG ∗ ℤ)NZD(DG ∗ ℤ)^{−1}

exists, is semisimple, and is as always universal NZD(DG ∗ ℤ)-inverting, it is not clear that there is a map from this localization to UΓ. Our next aim is therefore to prove the equality T(DG ∗ ℤ ⊂ UΓ) = NZD(DG ∗ ℤ). At this point we have to use more than abstract ring theoretical arguments and investigate the situation again in terms of functional analysis. The key point is the following theorem of Linnell.

Theorem 8.15. Let 1 → G → Γ → ℤ → 1 be an exact sequence of groups. Choose a section μ : ℤ → Γ which is a group homomorphism. Set z = μ(1). Suppose the first non-vanishing coefficient of the Laurent polynomial f(z) ∈ N G ∗ ℤ is a non-zerodivisor in N G, then f(z) is a non-zerodivisor in N Γ.

Proof. Theorem 4 in [47] proves a more general statement.

This enables us to prove that non-zerodivisors in UG ∗ ℤ of a very special kind become invertible in UΓ.

Corollary 8.16. Suppose f(z) ∈ UG ∗ ℤ is of the form f(z) = 1 + a_{1}z + … + a_{n}z^{n}, then it is invertible in UΓ.

Proof. Remember that UG is a left and a right ring of fractions of N G (Proposition 2.8). Since there is a common denominator for the a_{1}, …, a_{n} we can find an s ∈ T(N G ⊂ UG) = NZD(N G) and a g(z) ∈ N G ∗ ℤ such that f(z) = s^{−1}g(z). The first non-vanishing coefficient of g(z) is s and the above theorem applies. We get g(z) ∈ NZD(N Γ) = T(N Γ ⊂ UΓ) ⊂ UΓ^{×} and therefore f(z) = s^{−1}g(z) ∈ UΓ^{×}.

Now we have to use the structure theory of semisimple rings to deduce the desired result. This is done in several steps.

Proposition 8.17. Suppose R ∗ ℤ ⊂ S is a crossed product and one knows that polynomials of the form f(z) = 1 + a_{1}z + … + a_{n}z^{n} ∈ R ∗ ℤ become invertible in S. Then if

(i) R is a skew field,

(ii) R is a simple artinian ring,

(iii) R is a semisimple ring,

then we have NZD(R ∗ ℤ) = T(R ∗ ℤ ⊂ S), and the Ore localization

(R ∗ ℤ)NZD(R ∗ ℤ)^{−1}

embeds into S as the division closure

D(R ∗ ℤ ⊂ S).

Proof. We have already discussed above in 8.12 that R∗ℤ with R semisimple is noetherian and semiprime. So indeed Goldie’s theorem applies, and once we know that every non-zerodivisor becomes invertible in S the Ore localization embeds into S as the division closure by Proposition 13.17(ii). In fact, with little effort one could improve the proof given below and reprove Goldie’s theorem in this special case.

(i) One verifies that up to a unit in R and multiplication by z^{n} every nonzero element of R ∗ ℤ can be written as f(z) above. The Ore localization in this case is a skew field. (ii) Step1: We will first change the crossed product structure on the ring R ∗ ℤ to obtain one with better properties. Let R = M_{n}(ℤ) ⊗_{ℤ} D with D a skew field be a simple artinian ring. Let μ denote the crossed product structure map for the ring R ∗ ℤ. Every automorphism c : R → R splits into an inner automorphism and an automorphism of the form

id ⊗ θ : M_{n}(ℤ) ⊗_{ℤ} D → M_{n}(ℤ) ⊗_{ℤ} D,

where θ is an automorphism of D, compare [38, page 237]. In particular c_{μ(1)} = c_{u} ∘ (id ⊗ θ) for some invertible element u ∈ R = M_{n}(ℤ) ⊗_{ℤ} D. If we now set μ̃(1) = u^{−1}μ(1) and μ̃(n) = (μ̃(1))^{n} we get a new crossed product structure map μ̃ such that

c_{μ̃(n)} : M_{n}(ℤ) ⊗_{ℤ} D → M_{n}(ℤ) ⊗_{ℤ} D

restricts to

c_{μ̃(n)} : ℤ · 1_{n} ⊗_{ℤ} D → ℤ · 1_{n} ⊗_{ℤ} D

and the corresponding twisting τ̃ is identically 1. We see that with respect to this new crossed product structure the subring (or left ℤ · 1_{n} ⊗_{ℤ} D-module) generated by ℤ · 1_{n} ⊗_{ℤ} D and μ̃(ℤ) is a sub-crossed product of (R ∗ ℤ, μ̃), i.e. the inclusion is a crossed product homomorphism. We denote this ring by ((ℤ · 1_{n} ⊗_{ℤ} D) ∗ ℤ, μ̃). Note that the assumption about polynomials is invariant under this base change since μ(n) and μ̃(n) differ by an invertible element in R. There is a natural isomorphism

M_{n}(ℤ) ⊗_{ℤ} ((ℤ · 1_{n} ⊗_{ℤ} D) ∗ ℤ) → (M_{n}(ℤ) ⊗_{ℤ} D) ∗ ℤ.

From now on we will write D instead of ℤ · 1_{n} ⊗_{ℤ} D, and via the above isomorphism we consider

M_{n}(ℤ) ⊗_{ℤ} (D ∗ ℤ) ⊂ S

as a subring of S. Step2: If we apply (i) to the inclusion ℤ · 1_{n} ⊗_{ℤ} (D ∗ ℤ) ⊂ S we see that ℤ · 1_{n} ⊗_{ℤ} (D ∗ ℤ) fulfills the Ore condition with respect to the set T of all nonzero elements. The Ore localization is a skew field K and embeds into S as the division closure

K = (ℤ · 1_{n} ⊗_{ℤ} (D ∗ ℤ)) T^{−1} = D (ℤ · 1_{n} ⊗_{ℤ} (D ∗ ℤ) ⊂ S) ⊂ S.

Since K is an Ore localization the following diagram is a push-out diagram by Corollary 13.8.

[figure: ℤ · 1_{n} ⊗_{ℤ} (D ∗ ℤ); ℤ · 1_{n} ⊗_{ℤ} K; M_{n}(ℤ) ⊗_{ℤ} (D ∗ ℤ); M_{n}(ℤ) ⊗_{ℤ} K.]

Since ℤ · 1_{n} ⊗_{ℤ} K embeds into S and M_{n}(ℤ) ⊗_{ℤ} (D ∗ ℤ) embeds into S there is an induced homomorphism

Φ : M_{n}(ℤ) ⊗_{ℤ} K → S.

Since D(M_{n}(ℤ)⊗_{ℤ}(D ∗ℤ) ⊂ S) contains M_{n}(ℤ)⊗_{ℤ}(D ∗ℤ) and ℤ·1_{n} ⊗_{ℤ}K it must contain im(Φ). Since im(Φ) is a homomorphic image of a the semisimple ring M_{n}(ℤ) ⊗_{ℤ} K it is semisimple and therefore division closed. We get equality. We claim that Φ is injective. The entries of a matrix in M_{n}(ℤ)⊗_{ℤ}K can always be brought to a common denominator. Injectivity follows from the above pushout diagram and the fact that the M_{n}(ℤ) ⊗_{ℤ} (D ∗ ℤ) → S is injective. So we have that

M_{n}(ℤ) ⊗_{ℤ} K ≅ D (M_{n}(ℤ) ⊗_{ℤ} (D ∗ ℤ) ⊂ S) ⊂ S.

Remains to be proven that all non-zerodivisors in M_{n}(ℤ) ⊗_{ℤ} (D ∗ ℤ) become invertible in S. Let a be an element in M_{n}(ℤ) ⊗ ℤ(D ∗ ℤ). Now a either becomes a zerodivisor or invertible in the semisimple ring M_{n}(ℤ)⊗_{ℤ}K. From Proposition 13.7 we know that

M_{n}(ℤ) ⊗_{ℤ} K ≅ (M_{n}(ℤ) ⊗_{ℤ} (D ∗ ℤ)) (T · 1_{n})^{−1}

is an Ore localization. Using this we verify that a must have been a zerodivisor from the beginning if it becomes a zerodivisor in M_{n}(ℤ) ⊗_{ℤ} K. (iii) Let 1 = ∑ e_{i} be a decomposition of the unit in R into central primitive orthogonal idempotents. Every automorphism of R permutes the e_{i}. Therefore we can find an m ∈ ℕ with c_{μ(1)}^{m}(e_{i}) = e_{i}, and the product structure induced on the subring R ∗ (mℤ) ⊂ R ∗ ℤ respects the decomposition of R into simple artinian rings, i.e. the e_{i} are still central orthogonal idempotents for R ∗ mℤ

R ∗ mℤ = (⊕_{i}Re_{i}) ∗ mℤ = ⊕_{i}(R ∗ mℤ)e_{i}.

The hypothesis implies that every f(z) ∈ (R ∗ mℤ)e_{i} with leading term e_{i} is invertible in e_{i}Se_{i}. From (ii) we therefore get

NZD(R ∗ mℤe_{i}) = T(R ∗ mℤe_{i} ⊂ e_{i}Se_{i}).

One checks that an element f ∈ R ∗ mℤ is invertible in e_{i}Se_{i} if and only if every fe_{i} is invertible in e_{i}Se_{i}, and it is a non-zerodivisor in R ∗ mℤ if and only if all fe_{i} are non-zerodivisors in R ∗ mℤe_{i}. Therefore

NZD(R ∗ mℤ) = T(R ∗ mℤ ⊂ S).

Goldie’s Theorem yields that

D(R ∗ mℤ ⊂ S) = (R ∗ mℤ)T(R ∗ mℤ ⊂ S)^{−1}

is semisimple. Note that R∗ℤ = (R∗mℤ)∗ℤ/mℤ and D(R∗mℤ ⊂ S)∗ℤ/mℤ is semisimple by 8.12 (iv) and therefore division closed. So T(R ∗ ℤ ⊂ S) = T(R ∗ ℤ ⊂ D(R ∗ mℤ ⊂ S) ∗ ℤ/mℤ). By 8.5 (vi) we have that

(R ∗ mℤ) ∗ ℤ/mℤ ⊂ D(R ∗ mℤ ⊂ S) ∗ ℤ/mℤ
= (R ∗ mℤ)T(R ∗ mℤ ⊂ S)^{−1} ∗ ℤ/mℤ
= ((R ∗ mℤ) ∗ ℤ/mℤ)T(R ∗ mℤ ⊂ S)^{−1}

is universal T(R ∗ mℤ ⊂ S)-inverting. Since it is also T(R ∗ ℤ ⊂ S)-inverting it is universal T(R ∗ ℤ ⊂ S)-inverting because one easily checks the corresponding universal property, compare 8.9(i). Therefore it is isomorphic to the Ore localization given by Goldie’s theorem.

It would be very interesting to have a similar statement if R is von Neumann regular, but of course the proof given here relies heavily on the structure theory for semisimple rings. Combining the above results we get

AIIb) (A)_{Y} ⇒ (A)_{Y{infinite cyclic}}

CIIb) (C)_{Y} ⇒ (C)_{Y{infinite cyclic}}.

Or in words:

Proposition 8.18 ( AIIb) and CIIb) ). Let 1 → G → Γ → ℤ → 1 be an exact sequence of groups. If DG is semisimple and if ℂG ⊂ DG is universal Σ(G)-inverting, then

DΓ = (DG ∗ ℤ)T(DG ∗ ℤ ⊂ UΓ)^{−1}

is semisimple and ℂΓ ⊂ DΓ is universal Σ(Γ)-inverting. If moreover T(G) = NZD(ℂG) and if DG ≅ (ℂG)T(G)^{−1} is a classical ring of fractions for G, then T(Γ) = NZD(ℂΓ) and

DΓ ≅ (ℂΓ)T(Γ)^{−1}

is a classical ring of quotients for ℂΓ.

Proof. From the above discussion we know that DG∗ℤ is semiprime noetherian and T(DG ∗ ℤ ⊂ UΓ) = NZD(DG ∗ ℤ). Therefore the Ore localization

(DG ∗ ℤ)T(DG ∗ ℤ ⊂ UΓ)^{−1}

exists, is semisimple according to Goldie’s Theorem and isomorphic to the division closure D(DG ∗ ℤ ⊂ UΓ) = DΓ. That ℂΓ ⊂ DΓ is universal Σ(Γ)- inverting now follows from 8.10 (ii), where we identify DG and DΓ with RG and RΓ by 8.10 (iii). The last statement is 8.10 (iv).

One extension by a finitely generated abelian-by-finite group can always be replaced by finitely many extensions by an infinite cyclic group followed by one extension by a finite group. If we now combine AIIa) and AIIb) and similarly CIIa) and CIIb) we immediately get the following:

AIIa)+b) (A)_{Y} ⇒ (A)_{Y{f.g. abelian by finite}}

CIIa)+b) (C)_{Y} ⇒ (C)_{Y{f.g. abelian by finite}}.

The statement (A) only cares about the ring extension ℂΓ ⊂ DΓ. But we have also obtained information about the intermediate extension DG ∗ H ⊂ DΓ which we will need again in the (B)-part. We record a corresponding statement in the following Lemma.

Lemma 8.19. Let 1 → G → Γ → H → 1 be an exact sequence of groups with H finitely generated abelian-by-finite.

If DG is semisimple, then

T(DG ∗ H ⊂ UΓ) = NZD(DG ∗ H),

the pair (DG ∗ H, T(DG ∗ H ⊂ UΓ)) satisfies the Ore condition,

DΓ ≅ (DG ∗ H)T(DG ∗ H ⊂ UΓ)^{−1}

is a classical ring of fractions for DG ∗ H and semisimple.

Proof. In the proof of AIIb) we have already seen the corresponding statement for H = ℤ. Since an extension by a finitely generated abelian-by-finite group can be replaced by finitely many extensions by infinite cyclic groups followed by an extension by a finite group it remains to be verified that we can iterate. Apply 8.7 (iv) and 8.5 (vi).

8.2 Induction Step: Extensions - The (B)-Part

We now turn to the (B)-part of the induction step. Remember that our task is to prove the surjectivity of the map colim_{K∈Fin(Γ)}K_{0}(ℂK) → K_{0}(DΓ). If we combine 8.19 with the following proposition we can handle the passage from DG∗H to DΓ if H is finitely generated abelian-by-finite. Remember that for a ring R the abelian group G_{0}(R) is the free abelian group on isomorphism classes of finitely generated modules modulo the usual relations which come from exact sequences.

Proposition 8.20. (i) Suppose that T ⊂ R is a set of non-zerodivisors and that the pair (R, T) satisfies the right Ore condition, then the map R → RT^{−1} induces a surjection

G_{0}(R) → G_{0}(RT^{−1}).

(ii) If the ring R is semisimple, then the natural map K_{0}(R) → G_{0}(R) is an isomorphism.

Proof. (i) Let M be a finitely generated RT^{−1}-module. We have to find a finitely generated R-module N with N ⊗_{R} RT^{−1} ≅ M. Choose an epimorphism p : (RT^{−1})^{n} → M. Let i : R^{n} → (RT^{−1})^{n} be the natural map. Now N = p ∘ i(R^{n}) is a finitely generated R-submodule of M. Let j : N → M denote the inclusion. Now apply − ⊗_{R} RT^{−1} to the diagram

[figure: R^{n}; N; (RT^{−1})^{n}; i; M.; j]

Since localizing is exact j ⊗ id_{RT^{−1}} is injective. For every RT^{−1}-module L we have that L ⊗_{R} RT^{−1} ≅ L. In particular i ⊗ id_{RT^{−1}} is an isomorphism and therefore also j⊗_{RT^{−1}} is an isomorphism and N ⊗_{R} RT^{−1} ≅ M. (ii) Over a semisimple ring all modules are projective.

Our target DΓ as well as our sources ℂK (where K is finite) are semisimple rings, and thus we can pass to G-theory instead of K-theory during the proof.

Proposition 8.21. Suppose DG is semisimple and H is finitely generated abelian-by-finite, then the map

G_{0}(DG ∗ H) → G_{0}(DΓ) = K_{0}(DΓ)

is surjective.

Proof. Combine 8.19 and 8.20.

Before we go on let us explain why the naive approach of proving BIIa) and BIIb) separately would fail. In fact it is possible to prove a statement like

BIIb) (A)_{Y} and (B)_{Y} ⇒ (B)_{Y{infinite cyclic}}.

Given an extension 1 → G → Γ → ℤ → 1, every finite subgroup of Γ already lies in the subgroup G, and we are left with the task of proving that the map G_{0}(DG) → G_{0}(DG ∗ ℤ) is surjective. This is possible. Unfortunately extensions by finite groups are much more complicated. The map G_{0}(DG) → G_{0}(DG ∗ H) is of course in general not surjective if H is finite and G_{0}(DG ∗ H) really depends on the crossed product structure (take for example the trivial group for G). The ring theoretical structure of DG and the abstract knowledge of H is not sufficient. Fortunately we have the following theorem.

Theorem 8.22 (Moody’s Induction Theorem). Let R be a noetherian ring and H be a group which is a finite extension of a finitely generated abelian group. Then the natural map

colim_{K∈Fin(H)}G_{0}(R ∗ K) → G_{0}(R ∗ H)

is surjective.

Proof. This is due to Moody [60] and [61]. See also [13], [26] and Chapter 8 of [70].

In our situation we get in particular.

Proposition 8.23. Let 1 → G → Γ → H → 1 be an exact sequence with H finitely generated abelian-by-finite. Suppose DG is semisimple, then the map

colim_{L∈Fin(H)}G_{0}(DG ∗ L) → G_{0}(DG ∗ H)

is surjective.

Combining this with the induction hypothesis we finally get:

BIIa)+b) (Y{finite} = Y, (A)_{Y} and (B)_{Y}) ⇒ (B)_{Y{f.g. abelian-by-finite}}.

In words:

Proposition 8.24 ( BIIa)+b) ). Let Y be a class of groups which is closed under finite extensions. Let 1 → G → Γ → H → 1 be an exact sequence with G ∈ Y and let H be finitely generated abelian-by-finite. Suppose DG is semisimple and we know that

colim_{L∈FinG}K_{0}(ℂL) → K_{0}(DG)

is surjective for all groups G in Y. Then the map

colim_{K∈FinΓ}K_{0}(ℂK) → K_{0}(DΓ)

is surjective.

Proof. Let p : Γ → H be the quotient map. All maps in the following diagram are either isomorphisms or surjective.

colim_{M∈FinΓ}K_{0}(ℂM) →^{≅} colim_{K∈FinH}colim_{L∈Finp^{−1}(K)}K_{0}(ℂL) (1)
→ colim_{K∈FinH}K_{0}(Dp^{−1}(K))  (2)
→^{≅} colim_{K∈FinH}G_{0}(Dp^{−1}(K))  (3)
→^{≅} colim_{K∈FinH}G_{0}(DG ∗ K)  (4)
→ G_{0}(DG ∗ H)  (5)
→ G_{0}(DΓ)  (6)
→^{≅} K_{0}(DΓ)  (7)

We start at the bottom. (7) is an isomorphism by 8.20 (ii) since we know from AIIa)+b) that DΓ is semisimple. Map (6) is surjective by 8.21, map (5) by Moody’s induction theorem. (4) Since K is finite and DG semisimple we know from AIIa) that Dp^{−1}(K) = DG ∗ K. (3) This follows from 8.20 (ii). (2) By assumption the class Y is closed under extension by finite groups. Therefore p^{−1}(K) is again in Y and we know that

colim_{L∈Finp^{−1}(K)}K_{0}(ℂL) → K_{0}(Dp^{−1}(K))

is surjective. Since every M ∈ FinΓ occurs as M ∈ Fin(p^{−1}(p(M))) with p(M) ∈ FinH we get (1).

8.3 Induction Step: Directed Unions

This step is much easier than the preceding one. But the extra hypothesis about the orders of finite subgroups enters the proof. First we collect what we have to know about directed unions and localizations.

Proposition 8.25 (Directed Unions and Localization). Let R_{i} ⊂ S_{i} ⊂ S be subrings for all i ∈ I. Let R = ⋃_{i∈I} R_{i} and ⋃_{i∈I} S_{i} be directed unions. Suppose all the rings S_{i} are von Neumann regular.

(i) A directed union of von Neumann regular rings is von Neumann regular.

(ii) We have

T(R ⊂ S) = ⋃_{i∈I} T(R_{i} ⊂ S_{i})
Σ(R ⊂ S) = ⋃_{i∈I} Σ(R_{i} ⊂ S_{i}),

and the unions are directed.

(iii) We have

D(R ⊂ S) = ⋃_{i∈I} D(R_{i} ⊂ S_{i})
R(R ⊂ S) = ⋃_{i∈I} R(R_{i} ⊂ S_{i}),

and the unions are directed.

(iv) If all the D(R_{i} ⊂ S_{i}) are universal T(R_{i} ⊂ S_{i})-inverting, then D(R ⊂ S) is universal T(S)-inverting.

(v) If all the R(R_{i} ⊂ S_{i}) are universal Σ(R_{i} ⊂ S_{i})-inverting, then R(R ⊂ S) is universal Σ(S)-inverting.

(vi) Suppose all the pairs (R_{i}, T(R_{i} ⊂ S_{i})) satisfy the right Ore condition, then (R, T(R ⊂ S)) satisfies the right Ore condition.

(vii) If NZD(R_{i}) = T(R_{i} ⊂ S_{i}) for all i ∈ I, then NZD(R) = T(R ⊂ S).

(viii) If D(R_{i} ⊂ S_{i}) is a classical ring of right quotients for R_{i}, then D(R ⊂ S) is a classical ring of right quotients for R.

Proof. (i) This follows immediately if one makes use of the fact that a ring R is von Neumann regular if and only if for every x ∈ R there exists a y ∈ R such that xyx = x. (ii) Suppose x ∈ R becomes invertible in S, then since in the von Neumann regular ring S_{i} an element is either invertible or a zerodivisor (see 12.3(i)) it is already invertible in S_{i}. The other inclusion is clear. The argument for Σ(R ⊂ S) is similar using that matrix rings over von Neumann regular rings are again von Neumann regular.

(iii) Since S_{i} is von Neumann regular we have D(R_{i} ⊂ S_{i}) = D(R_{i} ⊂ S) ⊂ D(R ⊂ S) and therefore ⋃_{i∈I} D(R_{i} ⊂ S_{i}) ⊂ D(R ⊂ S). On the other hand ⋃_{i∈I} D(R_{i} ⊂ S_{i}) = ⋃_{i∈I} D(R_{i} ⊂ S) is division closed in S. Similar for the rational closure. (iv) Given a T(R ⊂ S)-inverting homomorphism R → R′. The composition R_{i} → R → R′ is T(R_{i} ⊂ S_{i})-inverting and the universal property gives a unique map ϕ_{i} : D(R_{i} ⊂ S_{i}) → R′. Using that the union is directed and each ϕ_{i} is unique one verifies that these maps determine a unique map D(R ⊂ S) = ⋃_{i∈I} D(R_{i} ⊂ S_{i}) → R′.

(v) is similar to (iv). (viii) follows from (vi) and (vii) which are easy.

If we apply this to our situation we get:

Proposition 8.26. If Γ = ⋃_{i∈I} Γ_{i} is a directed union of groups the following holds.

(i) The ring DΓ is the directed union of the subrings DΓ_{i} and RΓ is the directed union of the RΓ_{i}.

(ii) Suppose ℂΓ_{i} ⊂ DΓ_{i} is universal T(Γ_{i})-inverting for every i ∈ I. Then ℂΓ ⊂ DΓ is universal T(Γ)-inverting.

(iii) Suppose ℂΓ_{i} ⊂ RΓ_{i} is universal Σ(Γ_{i})-inverting for every i ∈ I. Then ℂΓ ⊂ RΓ is universal Σ(Γ)-inverting.

(iv) Suppose for all i ∈ I that NZD(ℂΓ_{i}) = T(Γ_{i}) and suppose DΓ_{i} is a classical ring of quotients for ℂΓ_{i}, then NZD(ℂΓ) = T(Γ) and DΓ is a classical ring of fractions for ℂΓ.

(v) If all the DΓ_{i} are von Neumann regular, then DΓ is von Neumann regular.

(vi) If all the RΓ_{i} are von Neumann regular, then RΓ is von Neumann regular.

We now start with the (B)-part.

BIII) (B)_{Y} ⇒ (B)_{LY}.

Somewhat more precisely:

Proposition 8.27 (BIII)). Let Γ be the directed union of the Γ_{i}, and suppose that all the maps colim_{K∈Fin(Γ_{i})}K_{0}(ℂK) → K_{0}(DΓ_{i}) are surjective. Then also the map

colim_{K∈Fin(Γ)}K_{0}(ℂK) → K_{0}(DΓ)

is surjective.

Proof. From the preceding proposition we know that DΓ is the directed union of the DΓ_{i}. Since K-theory is compatible with colimits and K ∈ FinΓ lies in some FinΓ_{i} the result follows.

We now turn to the (A)-part. In our situation the DΓ_{i} are semisimple and hence von Neumann regular. Note that the above proposition only yields that DΓ is von Neumann regular. We have already seen in example 5.11 that without the assumption on the orders of finite subgroups semisimplicity need not hold. But since we have already established the (B)-part we can apply 5.12 to get

AIII) (A)_{Y′} and (B)_{Y′} ⇒ (A)_{(LY)′}

CIII) (C)_{Y′} and (B)_{Y′} ⇒ (C)_{(LY)′}.

In words:

Proposition 8.28 (AIII) and CIII)). Let Γ be the directed union of the Γ_{i}. Suppose all the DΓ_{i} are semisimple and universal Σ(Γ_{i})-inverting. Suppose all the maps

colim_{K∈Fin(Γ_{i})}K_{0}(ℂK) → K_{0}(DΓ_{i})

are surjective. If there is a bound on the orders of finite subgroups of Γ, then DΓ is semisimple and universal Σ(Γ)-inverting. If moreover NZD(ℂΓ_{i}) = T(Γ_{i}) and all the pairs (ℂΓ_{i}, T(Γ_{i})) satisfy the Ore condition, then NZD(ℂΓ) = T(Γ) and the pair (ℂΓ, T(ℂΓ)) satisfies the Ore condition.

Proof. The first part has been explained above. The second part is 8.26 (iv).

8.4 Starting the Induction - Free Groups

We begin with the promised proof of Theorem 5.16. Because of the following we need not distinguish between the rational and the division closure in the case of a torsionfree group.

Note 8.29. The ring DΓ is a skew field if and only if RΓ is a skew field, and in this case the two rings coincide.

Proof. Suppose DΓ is a skew field. Then it is not only division closed but also rationally closed by 13.15 and therefore DΓ = RΓ is a skew field. Conversely, if RΓ is a skew field, then DΓ is a division closed subring of a skew field and therefore itself a skew field.

Proposition 8.30. Let Γ be a torsionfree group and let R ⊂ ℂ be a subring. The Atiyah conjecture with R-coefficients holds if and only if D(RΓ ⊂ UΓ) is a skew field.

Proof. Suppose DΓ = D(RΓ ⊂ UΓ) is a skew field. Let M be a finitely presented RΓ-module, then M ⊗_{RΓ} DΓ is a finite dimensional vector space and dim_{UΓ}(M ⊗_{RΓ} DΓ ⊗_{DΓ} UΓ) is an integer. On the other hand suppose the Atiyah conjecture holds. Let x be a nontrivial element in the rational closure RΓ = R(RΓ ⊂ UΓ). We will show that x is invertible in UΓ. Since RΓ is rationally closed and in particular division closed the inverse x^{−1} lies in RΓ and we see that this ring is a skew field. By the note above also DΓ is a skew field. Let 0 ≠ x ∈ RΓ. The following lemma tells us that x is stably associated over RΓ (over UΓ would be sufficient) to a matrix A over RΓ. Interpreting all matrices as matrices over UΓ we get an isomorphism of UΓ-modules im(x) ⊕ ⨁_{i=1}^{n} UΓ ≅ im(A). By the Atiyah conjecture dim(im(A)) has to be an integer and therefore dim im(x) is either 0 or 1. Additivity of the dimension and von Neumann regularity of UΓ implies that x is either 0 or invertible.

Lemma 8.31. Let R ⊂ S be a ring extension. Then every matrix over R(R ⊂ S) is stably associated over R(R ⊂ S) to a matrix over R.

Proof. We deduce this from the corresponding statement for the universal localization. Let R → R_{Σ(R⊂S)} be universal Σ(R ⊂ S)-inverting. We know that the map R_{Σ(R⊂S)} → R(R ⊂ S) given by the universal property is surjective. So we lift the matrix to R_{Σ(R⊂S)} use 8.8 (i) and map everything back to R(R ⊂ S).

We have already seen that the Atiyah conjecture holds for the free group on two generators. Since the conjecture is stable under taking subgroups we get from the preceding proposition.

Proposition 8.32. If Γ is a free group, then DΓ is a skew field.

Proof. The group Γ is the directed union of its finitely generated subgroups ⋃_{i∈I} Γ_{i}. Every finitely generated free group is a subgroup of the free group on two generators. Since the Atiyah conjecture is stable under taking subgroups the preceding proposition yields, that all the DΓ_{i} are skew fields. By 8.26 DΓ = ⋃_{i∈I} DΓ_{i}. One easily verifies that a directed union of skew fields is again a skew field.

The hard part is now to verify that ℂΓ ⊂ DΓ is universal Σ(Γ)-inverting. Unfortunately we have to introduce a new concept: the universal field of fractions of a given ring R. This should not be confused with any of the other notions introduced so far (universal Σ-inverting ring homomorphism, universal T -inverting ring homomorphism, classical ring of fractions). As the name suggests, the universal field of fractions is defined by a universal property, but unfortunately it does not have to exist. For more details the reader is referred to Appendix III.

Proposition 8.33. Let R be a semifir, then the universal field of fractions R → K exists and is universal Σ(R → K)-inverting.

Proof. See Appendix III Proposition 13.20.

The group ring ℂΓ of a free group is the standard example of a fir [16][Section 10.9]. In particular it is a semifir. We still have to recognize DΓ as a universal field of fractions for ℂΓ. From Corollary 13.25 we know that it is sufficient to show that ℂΓ ⊂ DΓ is Hughes-free. By definition this means the following: Given any finitely generated subgroup G ⊂ Γ and a t ∈ G together with a homomorphism p_{t} : G → ℤ which maps t to a generator, the set {t^{i} | i ∈ ℤ} is D(ℂ ker p_{t} ⊂ UΓ)-left linearly independent. But we know from the exact sequence

1 → ker p_{t} → G → ℤ → 1

and Proposition 8.5(i) that the ring generated by

D(ℂ ker p_{t} ⊂ UΓ) = D(ℂ ker p_{t} ⊂ U ker p_{t}) = D(ker p_{t})

and G as a subring of U(ker p_{t}) ∗ ℤ is itself a crossed product D(ker p_{t}) ∗ ℤ. In particular it is a free D(ker p_{t})-module with basis {t^{i} | i ∈ ℤ}.

Proposition 8.34 (AIa)). If Γ is a free group, then DΓ is a skew field and ℂΓ ⊂ DΓ is universal Σ(Γ)-inverting.

Note that once we know DΓ is a skew field the (B)-part is trivially verified since K_{0}(DΓ) ≅ ℤ.

8.5 Starting the Induction - Finite Extensions of Free Groups

We now turn to the case of finite extensions of a free groups. We already know from AIIa) that extensions by finite groups cause no problems for the (A)-part.

Proposition 8.35 (AIb)). Let Γ be a finite extension of a free group, then DΓ is semisimple and ℂΓ ⊂ DΓ is universal Σ(Γ)-inverting.

The (B)-part remains to be proven. This time we really follow the philosophy outlined in Subsection 5.3 and begin with the statement that would follow from the isomorphism conjecture in algebraic K-theory.

Proposition 8.36. Let Γ be a finite extension of a free group, then the map

colim_{K∈Fin(Γ)}K_{0}(ℂK) → K_{0}(ℂΓ)

is surjective.

Proof. See [48][Lemma 4.8].

We still have to show surjectivity of the map K_{0}(ℂΓ) → K_{0}(DΓ). Since we already know that ℂΓ ⊂ DΓ is universal Σ(Γ)-inverting we can use results of Schofield [78] about the K-theory of universal localizations. Again we have to introduce a few new concepts which we discuss in Appendix III.

Proposition 8.37. Let R be a hereditary ring with faithful projective rank function ρ : K_{0}(R) → ℝ and let Σ be a set of full matrices with respect to ρ, then the universal Σ-inverting homomorphism R → R_{Σ} induces a surjection

K_{0}(R) → K_{0}(R_{Σ}).

Proof. This is a simplification of Theorem 5.2 of [78]. Note that by Lemma 1.1 and Theorem 1.11 in the same book the projective rank function is automatically a Sylvester projective rank function and by Theorem 1.16 there are enough left and right full maps.

Let us explain how this applies in our situation. Let Γ be finitely generated and let G be a free subgroup of finite index. Then Γ operates with finite isotropy on a tree, and the ℂΓ-module ℂ admits of a 1-dimensional projective resolution. Compare [21][Chapter IV, Corollary 3.16 on page 114]. This implies that every ℂΓ-module has a 1-dimensional projective resolution (see [77][Proposition 8.2.19 on page 315]), and in particular ℂΓ is hereditary. Now the dimension function for UΓ gives us a faithful projective rank function ρ = dim_{UΓ}(− ⊗_{ℂΓ} UΓ). We check in Lemma 13.30 that Σ(ℂΓ ⊂ UΓ) is a set of full maps with respect to ρ. Let us record the result.

Proposition 8.38. If Γ is a finite extension of a finitely generated free group, then the map K_{0}(ℂΓ) → K_{0}(DΓ) is surjective.

Since K-theory is compatible with colimits and DΓ is the directed union of the DΓ_{i} for the finitely generated subgroups Γ_{i} of Γ we finally get our result.

Proposition 8.39 (BIb)). If Γ is a finite extension of a free group, then the map

colim_{K∈Fin(Γ)}K_{0}(ℂK) → K_{0}(DΓ)

is surjective.

This finishes the proof of Theorem 8.3 and Theorem 8.4.

9 Homological Properties and Applications

In this section we will apply our careful investigation of the rings DΓ to obtain homological information about our rings. We have already mentioned that for non-amenable groups we cannot expect that the functor − ⊗_{ℂΓ} UΓ is exact. The following gives a precise result for groups in the class C.

Theorem 9.1. (i) Let Γ be in the class C with a bound on the orders of finite subgroups, then

Tor_{p}^{ℂΓ}(−; DΓ) = 0 for all p > 1.

(ii) If moreover Γ is elementary amenable, then

Tor_{p}^{ℂΓ}(−; DΓ) = 0 for all p > 0,

i.e. the functor − ⊗_{ℂΓ} DΓ is exact.

Note that for these groups DΓ is semisimple and therefore the functor −⊗_{DΓ} UΓ is exact. The functor − ⊗_{ℤΓ} ℂΓ is always exact. Therefore we immediately get the corresponding statements for Tor_{p}^{ℂΓ}(−; UΓ), Tor_{p}^{ℤΓ}(−; DΓ) and Tor_{p}^{ℤΓ}(−; UΓ). This has consequences for L^{2}-homology.

Corollary 9.2. Let Γ be in C with a bound on the orders of finite subgroups. Then there is a universal coefficient theorem for L^{2}-homology: Let X be a Γ-space whose isotropy groups are all finite, then there is an exact sequence

0 → H_{n}(X; ℤ) ⊗_{ℤΓ} UΓ → H_{n}^{Γ}(X; UΓ) → Tor_{1}(H_{n−1}(X; ℤ); UΓ) → 0.

Proof. If X has finite isotropy, then the set of singular simplices also has only finite isotropy groups. If H is a finite subgroup of Γ, then ℂ [Γ/H] ≅ ℂΓ ⊗_{ℂH} ℂ is induced from the projective ℂH-module ℂ and therefore projective. We see that the singular chain complex with complex coefficients C_{∗} = C_{∗}^{sing}(X; ℂ) is a complex of projective ℂΓ-modules. The E^{2}-term of the Künneth spectral sequence (compare Theorem 5.6.4 on page 143 in [85])

E_{pq}^{2} = Tor_{p}^{ℂΓ}(H_{q}(C_{∗}); DΓ) ⇒ H_{p+q}(C_{∗} ⊗ DΓ) = H_{p+q}(X; DΓ)

is concentrated in two columns. The spectral sequence collapses, and we get exact sequences

0 → H_{n}(X; ℂ) ⊗_{ℂΓ} DΓ → H_{n}(X; DΓ) → Tor_{1}^{ℂΓ}(H_{n−1}(X; ℂ); DΓ) → 0.

Applying the exact functor − ⊗_{DΓ} UΓ yields the result.

If we apply Theorem 9.1 to the trivial ℂΓ-module ℂ we get:

Corollary 9.3. If Γ belongs to C and has a bound on the orders of finite subgroups, then

H_{p}(Γ; UΓ) = Tor_{p}^{ℂΓ}(ℂ; UΓ) = 0 for all p ≥ 2.

In particular, if the group is infinite we have b_{0}^{(2)}(Γ) = 0 and therefore

χ^{(2)}(Γ) ≤ 0,

since b_{1}^{(2)}(Γ) is the only L^{2}-Betti number which could possibly be nonzero.

As we have discussed in Section 5, the L^{(2)}-Euler characteristic coincides with the virtual or the usual Euler-characteristic whenever these are defined. The proof of Theorem 9.1 depends on the following Lemma.

Lemma 9.4. (i) Let R∗G ⊂ S ∗G be a compatible ring extension. Let M be an R∗G-module. There is a natural isomorphism of right S-modules

Tor_{p}^{R∗G}(M; S ∗ G) ≅ Tor_{p}^{R}(res_{R}^{R∗G}M; S)

for all p ≥ 0.

(ii) Suppose R ⊂ S is a ring extension and R = ⋃_{i∈I} R_{i} is the directed union of the subrings R_{i}. Let M be an R-module. Then there is a natural isomorphism of right S-modules

Tor_{p}^{R}(M; S) ≅ colim_{i∈I}Tor_{p}^{R_{i}}(res_{R_{i}}^{R}M; S_{i}) ⊗_{S_{i}} S

for all p ≥ 0.

Proof. (i) We start with the case p = 0. As usual we denote the crossed product structure map by μ. Define a map

h_{M} : res_{R}^{R∗G}M ⊗_{R} S → M ⊗_{R∗G} S ∗ G

by m ⊗ s ↦ m ⊗ s. Obviously h is a natural transformation from the functor res_{R}^{R∗G}(−) ⊗_{R} S to − ⊗_{R∗G} S ∗ G. If M = R ∗ G the map h_{R∗G}^{−1} : R ∗ G ⊗_{R∗G} S ∗ G ≅ S ∗ G → res_{R}^{R∗G}R ∗ G ⊗_{R} S given by sμ(g) ↦ g ⊗ c_{g}^{−1}(s) is a well-defined inverse. Since h is compatible with direct sums we see that h_{F} is an isomorphism for all free modules F. Now if M is an arbitrary module choose a free resolution F_{∗} → M of M and apply both functors to

F_{1} → F_{0} → M → 0 → 0.

Both functors are right exact, therefore an application of the five lemma yields the result for p = 0. Now let P_{∗} → M be a projective resolution of M, then

Tor_{p}^{R}(res_{R}^{R∗G}M; S) = H_{p}(res_{R}^{R∗G}P_{∗} ⊗_{R} S)
→^{≅} H_{p}(P_{∗} ⊗_{R∗G} S ∗ G)
= Tor_{p}^{R}(M; S ∗ G).

(ii) Again we start with the case p = 0. The natural surjections res_{R_{i}}^{R}M ⊗_{R_{i}} S → M ⊗_{R} S induce a surjective map

h_{M} : colim_{i∈I}res_{R_{i}}^{R}M ⊗_{R_{i}} S → M ⊗_{R} S

which is natural in M. Suppose the element of the colimit represented by ∑_{k} m_{k} ⊗ s_{k} ∈ res_{R_{i}}^{R}M ⊗_{R_{i}} S is mapped to zero in M ⊗_{R} S. By construction the tensor product M ⊗_{R} S is the quotient of the free module on the set M × S by a relation submodule. But every relation involves only finitely many elements of R, so we can find a j ∈ I such that ∑_{k} m_{k} ⊗ s_{k} = 0 already in res_{R_{j}}^{R}M ⊗_{R_{j}} S. We see that h_{M} is an isomorphism. Now let P_{∗} → M be a projective resolution. Since the colimit is an exact functor it commutes with homology and we get

colim_{i∈I}Tor_{p}^{R_{i}}(M; S) = colim_{i∈I}H_{p}(res_{R_{i}}^{R}P_{∗} ⊗_{R_{i}} S)
= H_{p}(colim_{i∈I}(res_{R_{i}}^{R}P_{∗} ⊗_{R_{i}} S))
→^{≅} H_{p}(P_{∗} ⊗_{R} S)
= Tor_{p}^{R}(M; S).

Proof of Theorem 9.1. (i) The proof works via transfinite induction over the group as explained in 7.9. (I) We first verify the statement for free groups. Let Γ be the free group generated by the set S. The cellular chain complex of the universal covering of the obvious 1-dimensional classifying space gives a projective resolution of the trivial module of length one

0 → ⨁_{S} ℂΓ → ℂΓ → ℂ → 0.

Now if M is an arbitrary ℂΓ-module we apply −⊗_{ℂ}M to the above complex and get a projective resolution of length 1 for M (diagonal action), compare Proposition 8.2.19 on page 315 in [77]. In particular we see that

Tor_{p}^{ℂΓ}(M; DΓ) = 0 for p > 1.

(II)” The next step is to prove that the statement remains true under extensions by finite groups. So let 1 → G → Γ → H → 1 be an exact sequence with H finite. From Proposition 8.13 we know that DΓ = DG ∗ H. Let M be a ℂΓ-module, then with Lemma 9.4 and the induction hypothesis we conclude

Tor_{p}^{ℂΓ}(M; DΓ) = Tor_{p}^{ℂG∗H}(M; DG ∗ H)
≅ Tor_{p}^{ℂG}(res_{ℂG}^{ℂG∗H}M; DG)
= 0 for p > 1.

The case H infinite cyclic is only slightly more complicated. This time we know from Proposition 8.18 (AIIb)) that DΓ = (DG ∗ H)T^{−1} is an Ore localization. Since Ore localization is an exact functor we get

Tor_{p}^{ℂΓ}(M; DΓ) = Tor_{p}^{ℂG∗H}(M; (DG ∗ H)T^{−1})
≅ Tor_{p}^{ℂG∗H}(M; DG ∗ H) ⊗_{DG∗H} DΓ

and conclude again with Lemma 9.4 that this module vanishes if p > 1. (III) The behaviour under directed unions remains to be checked. Let Γ = ⋃_{i∈I} Γ_{i} be a directed union, then we know from Proposition 8.26 that DΓ = ⋃_{i∈I} DΓ_{i}, and Lemma 9.4 gives

Tor_{p}^{ℂΓ}(M; DΓ) ≅ colim_{i∈I}Tor_{p}^{ℂΓ_{i}}(res_{ℂΓ_{i}}^{ℂΓ}M; DΓ)
= colim_{i∈I}Tor_{p}^{ℂΓ_{i}}(res_{ℂΓ_{i}}^{ℂΓ}M; DΓ_{i}) ⊗_{DΓ_{i}} DΓ
= 0 for p > 1.

(ii) The proof is exactly the same, except that this time we start with the trivial group and hence we can extend the vanishing results to p > 0, compare 7.3.

10 K-theory

In this section we collect information about the algebraic K-theory of our rings ℂΓ, DΓ, N Γ and UΓ. Moreover, we compute K_{0} of the category of finitely presented N Γ-torsion modules. Every von Neumann algebra can be decomposed into a direct sum of algebras of type I, II_{1}, II_{∞} and III, compare [82, Chapter V, Theorem 1.19]. Since a group von Neumann algebra is finite only the types I and II_{1} can occur. So a priori for a group Γ we have N Γ = N Γ_{I} ⊕ N Γ_{II_{1}}. Remember that a group is called virtually abelian if it contains an abelian subgroup of finite index.

Proposition 10.1. Let Γ be a finitely generated group.

(i) If Γ is virtually abelian, then N Γ = N Γ_{I} is of type I and can be written as a finite direct sum

N Γ = ⊕_{n=1}^{k}N Γ_{I_{n}},

where N Γ_{I_{n}} is an algebra of type I_{n}.

(ii) If Γ is not virtually abelian, then N Γ = N Γ_{II_{1}} is of type II_{1}.

If Γ is not finitely generated mixed types can occur.

Proof. See page 122 in [36] and the references given there.

A von Neumann algebra of type I_{n} is isomorphic to M_{n}(Ł^{∞}(X; μ)) for some measure space (X, μ). Note that K-theory is Morita invariant and compatible with direct sums.

10.1 K_{0} and K_{1} of A and U

Let us start with K_{0}. As usual we fix a normalized trace tr_{A} on A. We denote the center of A by Z(A). The center valued trace for A is a linear map

tr_{Z(A)} : A → Z(A),

which is uniquely determined by tr_{Z(A)}(ab) = tr_{Z(A)}(ba) for all a, b ∈ A and tr_{Z(A)}(c) = c for all c ∈ Z(A), see [41, Proposition 8.2.8, p.517]. Our fixed trace tr_{A} : A → ℝ factorizes over the center valued trace, i.e. we have tr_{A}(a) = tr_{A}(tr_{Z(A)}(a)) for all a ∈ A, compare [41, Proposition 8.3.10, p.525].

Proposition 10.2. Let A be a finite von Neumann algebra with center Z(A) and let U be the associated algebra of affiliated operators. Let Z(A)_{sa} denote the vector space of selfadjoint elements in Z(A) and let Z(A)_{pos} denote the cone of nonnegative elements. Remember that Proj(A) denotes the monoid of isomorphism classes of finitely generated projective A-modules.

(i) The natural map A → U induces an isomorphism

K_{0}(A) \overline{≅^{}}K_{0}(U).

(ii) There is a commutative diagram

Proj(A) \overline{}K_{0}(A)
[figure: dim_{A}]
[figure: Z(A)_{pos}; Z(A)_{sa}; dim_{Z(A)}; ℝ.]

The map dim_{Z(A)} is induced by the center valued trace. All maps in the square are injective.

(iii) If A is of type II_{1}, then dim_{Z(A)} is an isomorphism.

(iv) If A = M_{n}(Ł^{∞}(X; μ)), then the image of dim_{Z(A)} consists of all measurable step functions on X with values in \frac{1}{n}ℤ, where we identify the center of A with Ł^{∞}(X; μ).

Proof. (i) This has already been proven in Theorem 3.8. (ii) In Section 3 we normalized the traces tr_{M_{n}(A)} such that tr_{M_{n}(A)}(1_{n}) = n. We extend the center valued trace to matrices by setting

tr_{n} : M_{n}(A) → Z(A), (a_{ij}) ↦ ∑ tr_{Z(A)}(a_{ii}).

These maps are compatible with the stabilization maps M_{n}(A) → M_{n+1}(A). Since \frac{1}{n}tr_{n} has the characteristic properties of a center valued trace if we identify Z(A) with Z(M_{n}(A)) we have

tr_{n} = n · tr_{Z(M_{n}(A))}.

Two projections p and q are Murray von Neumann equivalent and therefore isomorphic (compare Proposition 3.5) if and only if tr_{Z(A)}(p) = tr_{Z(A)}(q). The center valued trace of a projection is nonnegative and selfadjoint, see [41, Theorem 8.4.3, p.532]. Therefore we get injective maps

L_{P roj}(M_{n}(A))/≅ → Z(A)_{pos}

which are compatible with the stabilization maps. This gives an injective map Proj(A) → Z(A)_{pos}. Since the block sum of projections corresponds to an ordinary sum of traces we see that it is a map of monoids and that Proj(A) satisfies cancellation. Therefore the map to Proj(A) → K_{0}(A) is injective. This implies that also the induced map dim_{Z(A)} : K_{0}(A) → Z(A)_{+} is injective. (iii) By [41, Theorem 8.4.4 (i), p.533] we know that the image of the center valued trace in the case of a type II_{1} algebra is the set

{a ∈ Z(A) | a selfadjoint, nonnegative, |a| < 1}.

Because of the factor n in tr_{n} we get the result. (iii) This follows from [41, Theorem 8.4.4 (ii), p.533].

In [56] the algebraic K_{1}-groups of von Neumann algebras are determined. Of course they depend on the type of the von Neumann algebra. Moreover, the authors compute a modified K_{1}-group K_{1}^{w}(A). Let us recall its definition.

Definition 10.3. Let R be a ring. Let K_{1}^{inj}(R) be the abelian group generated by conjugation classes of injective endomorphisms of finitely generated free R-modules modulo the following relations:

(i) [f] = [g]+[h] if there is a short exact ladder of injective homomorphisms

[figure: 0; L; M; N; 0; 0; L; g; M; f; N; h; 0.]

(ii) [f ∘ g] = [f] + [g] if f and g are endomorphisms of the same module.

(iii) [id_{M}] = 0 for all modules M.

If R = A is a von Neumann algebra, then we define K_{1}^{w}(A) = K_{1}^{inj}(A).

Here w stands for weak isomorphism, compare Lemma 2.6. Note that if one replaces the word injective endomorphism by isomorphism one gets a possible definition of K_{1}(R). It turns out that the group K_{1}^{w}(A) admits a very natural interpretation.

Proposition 10.4. There is a natural isomorphism

K_{1}^{w}(A) \overline{≅^{}}K_{1}(U).

Proof. If we apply Lemma 2.6 to matrix algebras we see, that an endomorphism f between finitely generated free A-modules is injective if and only if f ⊗ id_{U} is an isomorphism. Therefore f ↦ [f ⊗ id_{U}] gives a well-defined map since the relations analogous to (i)–(iii) hold in K_{1}(U). To define an inverse we change the point of view and consider K_{1}(U) as GL(U)_{ab}. Let C be an invertible matrix over U. Let T ⊂ A be the set of non-zerodivisors in A. From Proposition 13.7 we know that M_{n}(U) = M_{n}(A)(T · 1_{n})^{−1} and therefore we can find a matrix A over A and s ∈ A such that C = As^{−1}1_{n}. Note that A and s1_{n} have to be injective endomorphisms because they become invertible over U. We want to show that the map

C ↦ [A] − [s1_{n}]

is well-defined. Suppose C = Bt^{−1}1_{n} is another decomposition of the matrix. The defining equivalence relation for the Ore localization tells us that there must exist matrices u1_{n} and v1_{n} over R such that Au1_{n} = Bv1_{n}, s1_{n}u1_{n} = t1_{n}v1_{n} and s1_{n}u1_{n} = t1_{n}v1_{n} ∈ T · 1_{n}, compare 13.3. Since u and v have to be weak isomorphisms one verifies, using relation (ii), that the map is well-defined. Since K_{1}^{w}(A) is abelian the map factorizes over K_{1}(U). That the maps are mutually inverse follows from the relations (i)–(iii).

An invertible operator a ∈ GL_{n}(A) has a gap in the spectrum near zero, therefore we can define the Fuglede-Kadison determinant via the function calculus as

det_{F K}(a) = exp (\frac{1}{2}tr_{Z(A)} (log(a^{∗}a))) .

The results from [56] can now be rephrased as follows.

Proposition 10.5. Let A be a finite von Neumann algebra and let U be the associated algebra of affiliated operators.

(i) If A is of type II_{1}, then

K_{1}(U) = 0

and the Fuglede Kadison determinant gives an isomorphism

det_{F K} : K_{1}(A) \overline{≅_{}}Z(A)_{pos}^{×},

where Z(A)_{pos}^{×} denotes the group of positive invertible elements in the center of A.

(ii) If A = M_{n}(Ł^{∞}(X; μ)) is of type I_{n}, then U = M_{n}(Ł(X; μ)) and we have the following commutative diagram

K_{1}(A)  \overline{}K_{1}(U)

[figure: Ł^{∞}(X; μ)^{×}; det; Ł(X; μ)^{×},; det]

where the vertical maps are isomorphisms.

Proof. See [56].

Let R be a ring. Sending an invertible element a ∈ R^{×} to the class of the corresponding 1 × 1-matrix in K_{1}(R) gives a map

(R^{×})_{ab} → K_{1}(R).

Here (R^{×})_{ab} is the abelianized group of units.

Proposition 10.6. Let A be a finite von Neumann algebra and let U be the associated algebra of affiliated operators. The maps

(A^{×})_{ab} → K_{1}(A) and (U^{×})_{ab} → K_{1}(U)

are isomorphisms.

Proof. For the first map this is proven more generally for finite AW^{∗}-algebras in [35, Theorem 7]. Since we know that U is a unit-regular ring the statement for the second map follows from [58] and [33].

10.2 Localization Sequences

Localizations yield exact sequences in K-theory. We will use such a sequence to compute K_{0} of the category of finitely presented A-torsion modules. For commutative rings or central localization the exact sequence already appears in [2]. The case of an Ore localization is treated for example in [5]. Schofield was able to generalize the localization sequence to universal localizations [78]. Since ℂΓ ⊂ DΓ is a universal localization for groups in C with a bound on the order of finite subgroups we treat the localization sequence in this generality. Let R be a ring and let Σ be a set of maps between finitely generated projective R-modules. Let \overline{Σ} be the saturation of Σ, compare page 123.

Definition 10.7. Given an injective universal localization R → R_{Σ} we define T_{R→R_{Σ}} to be the full subcategory of R-modules which are cokernels of maps from \overline{Σ}.

In the case of an Ore localization this category has a different description.

Note 10.8. If RT^{−1} is an Ore localization, then the category T_{R→RT^{−1}} is equivalent to the category of finitely presented torsion modules of projective dimension 1.

Proof. Suppose M is a finitely presented torsion module of projective dimension 1. Then there is a short exact sequence 0 → P → R^{n} → M → 0 with P finitely generated projective. Since − ⊗_{R} RT^{−1} is exact and M ⊗_{R} RT^{−1} = 0 the map P → R^{n} is in \overline{Σ}. On the other hand let f : P → Q be a map in \overline{Σ}. Since R → R_{Σ} is injective f must be injective and therefore gives a 1-dimensional resolution of coker(f) by finitely generated projective Rmodules.

Proposition 10.9 (Schofield’s Localization Sequence). Let R be a ring and let Σ be a set of maps between finitely generated projective R-modules. Let R → R_{Σ} be a universal Σ-inverting ring homomorphism which is injective. Then there is an exact sequence of algebraic K-groups

K_{1}(R) → K_{1}(R_{Σ}) → K_{0}(T_{R→R_{Σ}}) → K_{0}(R) → K_{0}(R_{Σ}).

Proof. We will only explain the maps. For a proof see [78, Theorem 4.12]. The simpler case of an Ore localization is also treated in [5]. Let a be an invertible matrix over R_{Σ}. The equality given in Proposition 13.11 induces the equality

[a] = [b] − [s]

in K_{1}(R_{Σ}) with s ∈ Σ̂ and b a matrix over R. Note that since a is invertible also b is invertible over R_{Σ} and therefore b ∈ \overline{Σ}. Therefore it makes sense to define

[a] ↦ [coker(s)] − [coker(b)] .

This gives the map K_{1}(R_{Σ}) → K_{0}(T_{R→R_{Σ}}). Now suppose M = coker(f : P → Q) with f ∈ \overline{Σ}, then one defines

[M] ↦ [Q] − [P] .

This gives the map K_{0}(T_{R→R_{Σ}}) → K_{0}(R). The other maps are the natural maps induced by the ring homomorphism R → R_{Σ}.

More information on localization sequences in algebraic K-theory can be found in [29], [31], [9], [80], [86], [83] and [5]. If we apply this to a finite von Neumann algebra A we can compute K_{0} of the category of finitely presented A-torsion modules, compare Subsection 3.3.

Proposition 10.10. Let A be a finite von Neumann algebra and let U be the associated algebra of affiliated operators.

(i) The category T_{A→U} is equivalent to the abelian category of finitely presented A-torsion modules.

(ii) If A is of type II_{1}, then K_{0}(T_{A→U}) = 0.

(iii) If A = M_{n}(Ł^{∞}(X; μ)) is of type I_{n}, then

K_{0}(T_{A→U}) = Ł(X; μ)/Ł^{∞}(X; μ).

Proof. (i) Over the semihereditary ring A every finitely presented module has projective dimension 1. (ii) This follows from the localization sequence since K_{0}(A) → K_{0}(U) is an isomorphism by Theorem 3.8 and K_{0}(U) = 0 by Proposition 10.5(i). (iii) Similarly with Proposition 10.5(ii).

Of course we can also apply the above sequence to the ring extension ℂΓ ⊂ DΓ in the cases where we know that this is a universal localization.

Note 10.11. Let Γ be a group in the class C with a bound on the order of finite subgroups. Then there is an exact sequence

K_{1}(ℂΓ) → K_{1}(DΓ) → K_{0}(T_{ℂΓ→DΓ}) → K_{0}(ℂΓ) → K_{0}(DΓ) → 0.

Proof. We have proven in Section 8 that for these groups the map

colim_{K∈FinΓ}K_{0}(ℂK) → K_{0}(DΓ)

is surjective. This map factorizes over K_{0}(ℂΓ) and therefore the last map is surjective.

Examples show that in general the map K_{0}(ℂΓ) → K_{0}(DΓ) is not even rationally injective. We also have an example of a group Γ which is not virtually abelian with K_{1}(DΓ) ≠ 0.

11 Appendix I: Affiliated Operators

The aim of this section is to prove that on the set U of closed densely defined operators affiliated to the finite von Neumann algebra A there is a welldefined addition and multiplication which turns U into an algebra. This fact goes back to the first Rings of Operators paper of Murray and von Neumann from 1936 [63]. The proof given below essentially follows their proof. For convenience we first collect a few facts about unbounded operators and fix notation. Let H be a Hilbert space. An unbounded operator a : H ⊃ dom(a) → H will simply be denoted by a. If b is an extension of a we write a ⊂ b, i.e. dom(a) ⊂ dom(b) and the restriction of b to dom(a) coincides with a. Given a, its graph is Γ_{a} = {(x, a(x))|x ∈ dom(a)} ⊂ H ⊕ H. An operator is closed if its graph is closed in H ⊕ H. An operator is closable if the closure of its graph is the graph of an operator. This operator is the minimal closed extension of a and will be denoted by [a]. If a is densely defined a^{∗} can be defined as usual with domain dom(a^{∗}) = {x ∈ H| < a(.), x > is a continuous linear functional on dom(a)}. If one adds or composes unbounded operators the domains of definition usually shrink:

dom(a + b) = dom(a) ∩ dom(b) dom(ab) = b^{−1}(dom(a)).

Here is a set of rules for unbounded operators.

Proposition 11.1. Let a,b and c be (unbounded) operators.

(i) (a + b)c = ac + bc.

(ii) a(b + c) ⊃ ab + ac, and equality holds if and only if a is bounded.

(iii) (ab)c = a(bc).

(iv) (a + b) + c = a + (b + c).

(v) If a ⊂ b, then ac ⊂ bc and ca ⊂ cb.

(vi) If a ⊂ b and c ⊂ d, then a + c ⊂ b + d.

(vii) If a is densely defined, then a^{∗} exists and is closed but not necessarily densely defined.

(viii) If a is densely defined and closed, then a^{∗} exists and is densely defined and closed.

(ix) Let a be densely defined. In order to show that a is closable it suffices to prove a^{∗} to be densely defined, because then a^{∗∗} is the desired closed extension.

(x) Let a be densely defined. Every extension b ⊃ a is densely defined and b^{∗} ⊂ a^{∗}.

(xi) Let a, b and ba be densely defined, then a^{∗}b^{∗} ⊂ (ba)^{∗}, and equality holds if b is bounded.

(xii) Let a, b and a + b be densely defined, then a^{∗} + b^{∗} ⊂ (a + b)^{∗}.

A densely defined operator is called symmetric if s ⊂ s^{∗}, it is called selfadjoint if s = s^{∗}. Note that by 11.1.(vii) a selfadjoint operator is automatically closed (and densely defined by definition). If a is densely defined and closed, then a^{∗}a is selfadjoint and the closure of the graph of a : dom(a^{∗}a) → H coincides with the graph Γ_{a} of a. One says dom(a^{∗}a) is a core for a.

Lemma 11.2. If a is densely defined and closed, then the operator 1 + a^{∗}a has a bounded inverse.

Proof. See [40, Theorem 2.7.8, p.158].

Given an operator a one defines the resolvent set as the set of those complex numbers λ for which a bounded operator b (the resolvent) exists with b(λ − a) ⊂ (λ − a)b = id_{H}. The complement of the resolvent set is the spectrum spec(a) of the operator a. As for bounded selfadjoint operators there is a spectral decomposition and a function calculus.

Proposition 11.3 (Spectral Decomposition). Let s be a selfadjoint operator (not necessarily bounded). Then spec(s) ⊂ ℝ and there exists a unique projection valued measure, i.e. a map e^{s} : B_{spec(s)} → B(H) from the Borel-σ-algebra to B(H) with the following properties:

(i) For every Borel subset Ω ⊂ spec(s) the operator e^{s}(Ω) is a projection.

(ii) e^{s}(∅) = 0 and e^{s}(spec(s)) = id_{H}.

(iii) For Borel sets Ω and Ω′ we have e^{s}(Ω ∩ Ω′) = e^{s}(Ω)e^{s}(Ω′).

(iv) e^{s} is strongly σ-additive, i.e. for a countable collection of pairwise disjoint Borel sets (Ω_{n})_{n∈ℕ} with Ω = ⋃_{n∈ℕ} Ω_{n} we have

lim_{N→∞} e^{s}(⋃_{n=1}^{N} Ω_{n})x = e^{s}(Ω)x

for all x ∈ H.

(v) A vector x ∈ H is in the domain dom(s) if and only if

∫_{ℝ} |λ|^{2} < e^{s}(dλ)x, x > < ∞,

where dλ denotes the Lebesgue measure on ℝ.

(vi) For x ∈ dom(s) and arbitrary y ∈ H we have

< s(x), y > = ∫_{ℝ} λ < e^{s}(dλ)x, y > .

One often defines e_{λ}^{s} = e^{s}((−∞, λ]), and passing to a Stieltjes integral the last fact is symbolically written as

s = ∫_{−∞}^{∞} λde_{λ}^{s}

and called the spectral decomposition of s.

The representation of s as an integral can be used to define new operators which are functions of s.

Proposition 11.4 (Function Calculus). Given a selfadjoint operator s and a Borel measurable function f : spec(s) → ℝ∪{±∞} such that e^{s}(f^{−1}({±∞}) = 0, there exists a closed and densely defined operator f(s) with the following properties.

(i) The vector x ∈ H is in the domain dom(f(s)) if and only if

∫_{ℝ} |f(λ)|^{2} < e^{s}(dλ)x, x > < ∞.

(ii) For every x ∈ dom(f(s)) and arbitrary y ∈ H we have

< f(s)x, y > = ∫_{ℝ} f(λ) < e^{s}(dλ)x, y > .

The assignment f ↦ f(s) is an essential homomorphism, i.e. it is a homomorphism of complex algebras if we define sum and product of (unbounded) operators as the closure of the usual sum and product.

Proof. Compare [73, Section 5.3].

As an application one derives the polar decomposition for unbounded operators.

Proposition 11.5 (Polar Decomposition). Let a be densely defined and closed, then there is a unique decomposition

a = us,

where u is a partial isometry and s is a nonnegative selfadjoint operator. More precisely s = \sqrt{a^{∗}a} in the sense of function calculus, ker u = ker a, u^{∗}us = s, u^{∗}a = s and uu^{∗}a = a.

Proof. [73, p.218]

Let now A ⊂ B(H) be a von Neumann algebra. We assume that 1_{A} = id_{H}.

Definition 11.6. If a is an operator and b ∈ B(H) is a bounded operator we say that a and b commute if ba ⊂ ab. An operator a is affiliated to A if it commutes with all bounded operators from the commutant A′. A closed subspace K ⊂ H is called affiliated to A if the corresponding projection p_{K} belongs to A. Given a set M of operators we define the commutant M′ to be the set of all bounded operators commuting with all operators in M.

Note that there is no condition concerning the domain of definition of an affiliated operator. Given a selfadjoint operator s it turns out that {s}′ is a von Neumann algebra and that s is affiliated to the von Neumann algebra W^{∗}(s) = {s}′′. Similarly, if the closed and densely defined operator a is not selfadjoint W^{∗}(a) = ({a}′ ∩ {a^{∗}}′)′ is a von Neumann algebra (see [73, p.206]).

Proposition 11.7. Let s be a selfadjoint operator. All spectral projections e^{s}(Ω) lie in W^{∗}(s) and all operators f(s) obtained via the function calculus are affiliated to W^{∗}(s).

Note that if s is affiliated to the von Neumann algebra A, then A′ ⊂ {s}′ and therefore W^{∗}(s) ⊂ A. The following lemma facilitates to verify whether a given operator is affiliated.

Lemma 11.8. Let a be an operator and let M be a ∗-closed set of bounded operators such that the algebra generated by M is dense in A′ with respect to the weak topology. The following statements are equivalent.

(i) a is affiliated to A.

(ii) For all unitary operators u ∈ A′ we have ua ⊂ au.

(iii) For all unitary operators u ∈ A′ we have ua = au.

(iv) For all operators m ∈ M we have ma ⊂ am.

Proof. (iv) ⇒ (i): Since M is ∗-closed one can verify using 11.1(x) and 11.1(xi) that also A^{∗} commutes with all elements of M. Hence M ⊂ {a}′ ∩ {a^{∗}}′ and since {a}′ ∩ {a^{∗}}′ is a von Neumann algebra A′ ⊂ {a}′ ∩ {a^{∗}}′.

(i) ⇒ (ii), (i) ⇒ (iv) and (iii) ⇒ (ii) are obvious. Since every operator can be written as a linear combination of four unitary operators (see [82, Proposition 4.9, p.20]) (ii) is a special case of (iv). It remains to be shown that (ii) implies (iii). Suppose u is a unitary operator and we have ua ⊂ au and u^{−1}a ⊂ au^{−1}. With 11.1(iii) and 11.1(v) we conclude a ⊂ u^{−1}au and u^{−1}au ⊂ a and therefore a = u^{−1}au

In particular we see that in the case of a group von Neumann algebra N Γ an operator on l^{2}Γ is affiliated to N Γ if it commutes with the right action of Γ on l^{2}Γ. It is occasionally convenient to have the following reformulations of the fact that an operator is affiliated.

Proposition 11.9. Let A ⊂ B(H) be a von Neumann algebra. Let a be a closed densely defined operator on H. The following statements are equivalent.

(i) a is affiliated to A.

(ii) Let a = us be the polar decomposition, then u ∈ A and s ∈ U.

(iii) Let a = us be the polar decomposition, then u ∈ A and all spectral projections e^{s}(Ω) of the operator s are in A.

Proof. We only show that (i) implies (ii). Let a be affiliated to A. Let a = us be the polar decomposition. If v is a unitary operator in A′ then vav^{−1} = vuv^{−1}vsv^{−1}. Since vuv^{−1} is unitary and vsv^{−1} is selfadjoint it follows from the uniqueness of the polar decomposition that vuv^{−1} = u and vsv^{−1}.

We now come back to our main object of study.

Definition 11.10 (The Algebra of Affiliated Operators). Let A be a finite von Neumann algebra. The set U is defined as the set of all operators a with the following properties.

(i) a is densely defined.

(ii) a is closed.

(iii) a is affiliated to A.

As already mentioned, the aim of this section is to prove that there is a welldefined addition, multiplication and involution on U, such that U becomes a complex ∗-algebra that contains A as a ∗-subalgebra. The only obvious algebraic operation in U is the involution:

Proposition 11.11. If a ∈ U, then a^{∗} ∈ U.

Proof. According to 11.1.(viii) a^{∗} is closed, densely defined and for every b ∈ A′, ba ⊂ ab implies b^{∗}a^{∗} ⊂ (ab)^{∗} ⊂ (ba)^{∗} ⊂ a^{∗}b^{∗} and therefore a^{∗} is affiliated to A since A′ is ∗-closed.

To deal with the addition and composition one has to ensure that the domains remain dense. At this point it is crucial that the von Neumann algebra A is finite and one has a notion of dimension. So let us fix a trace tr. Remember that a closed linear subspace P ⊂ H is called affiliated to A if the projection p onto that subspace is affiliated to A. For closed affiliated subspaces there is a dimension given by dim(P) = tr(p).

Definition 11.12. Define for an arbitrary linear subspace L ⊂ H

dim(L) = sup{dim(P) | P ⊂ H is closed and affiliated}

A linear subspace L ⊂ H is called essentially dense if dim(L) = 1.

This notion of dimension extends the usual notion of dimension if P ⊂ H is closed and affiliated. One verifies that a subspace L is essentially dense if and only if there is a sequence P_{1} ⊂ P_{2} ⊂ P_{3} ⊂ … ⊂ L of closed affiliated subspaces, such that lim_{i→∞} dim P_{i} = dim H = 1. This follows from the monotony of the dimension for closed affiliated subspaces. The continuity property of the dimension implies that an essentially dense subset is dense.

Lemma 11.13. The intersection of countably many essentially dense subspaces is again essentially dense.

Proof. Let L_{α}, α = 1, 2, 3, … be a sequence of essentially dense subspaces and let L = ⋂ L_{α}. Let N > 0 be given. We will show that there exists a closed affiliated subspace P ⊂ L with dim(P) ≥ 1 − 1/2^{N}. Choose P_{α} ⊂ L_{α} such that dim P_{α}^{⊥} ≤ 1/2^{N+α}. We will show by induction that dim(⋂_{α≤n} P_{α})^{⊥} ≤ ∑_{α=1}^{n} 1/2^{N+α}. There is a decomposition into orthogonal subspaces ⋂_{α<n} P_{α} = [⋂_{α<n} P_{α} ∩ P_{n}] ⊕ [⋂_{α<n} P_{α} ∩ (P_{n}^{⊥})] which implies

dim ⋂_{α≤n} P_{α} = dim ⋂_{α<n} P_{α} − dim ⋂_{α<n} P_{α} ∩ (P_{n}^{⊥})

≥ (1 − ∑_{α=1}^{n−1} 1/2^{N+α}) − 1/2^{N+n}

= 1 − ∑_{α=1}^{n} 1/2^{N+α},

and hence

dim( ⋂_{α≤n} P_{α})^{⊥} ≤ ∑_{α=1}^{n} 1/2^{N+α} = 1/2^{N} ∑_{α=1}^{n} 1/2^{α} ≤ 1/2^{N}.

Note that ⋂_{α} P_{α} is closed and affiliated. Since (⋂_{α} P_{α})^{⊥} is the directed union of the (⋂_{α≤n} P_{α})^{⊥} we conclude from the continuity property of the dimension that dim(⋂_{α} P_{α})^{⊥} ≤ 1/2^{N}. Since ⋂_{α} P_{α} ⊂ ⋂_{α} L_{α} and N was chosen arbitrarily we see that dim ⋂_{α} L_{α} = 1.

Lemma 11.14. If a ∈ U and L ⊂ H is essentially dense, then a^{−1}(L) is essentially dense. Moreover dom(a) is essentially dense for every a ∈ U.

Proof. Using the polar decomposition a = us one can treat the cases a = u bounded and a = s selfadjoint separately. So let a be bounded. For every closed affiliated subspace P ⊂ L, the subspace a^{−1}(P) ⊂ a^{−1}(L) is closed since a is continuous and affiliated. (It is sufficient to ensure that u(a^{−1}(P)) ⊂ a^{−1}(P) for all unitary operators u from the commutant A′.). Moreover, dim(a^{−1}(P)) ≥ dim(P) and we see that dim(a^{−1}(L)) = 1. Now let a = s be selfadjoint and nonnegative. Let s = ∫ λde_{λ} be the spectral decomposition of s. Note that e_{λ}H ⊂ dom(s) and se_{λ} is a bounded operator. The argument above shows that given ϵ_{1} > 0 we can find a closed affiliated subspace P ⊂ L with dim(se_{λ})^{−1}(P) ≥ 1 − ϵ_{1}. On the other hand given any ϵ_{2} > 0 we can find λ such that dim(e_{λ}H)^{⊥} ≤ ϵ_{2}. Now the orthogonal decomposition (se_{λ})^{−1}(P) = (se_{λ})^{−1}(P)∩e_{λ}H ⊕(se_{λ})^{−1}(P)∩(e_{λ}H)^{⊥} implies that dim(se_{λ})^{−1}(P) ∩ e_{λ}H ≥ 1 − (ϵ_{1} + ϵ_{2}). Since (se_{λ})^{−1}(P) ∩ e_{λ}H is closed, affiliated and a subspace of s^{−1}(P) we see that dim(s^{−1}(P)) = 1. Let us now prove the second statement. If a = us is the polar decomposition, then dom(a) = dom(s). Since e_{λ} converges strongly to id_{H} for λ → ∞ we see that dim(dom(s)) = 1.

Lemma 11.15. If a ∈ U and b ∈ U, then a + b and ab are densely defined.

Proof. This follows immediately from the above lemmata.

Proposition 11.16. If a ∈ U and b ∈ U, then a + b and ab are affiliated and closable. Moreover the closures [a + b] and [ab] are in U.

Proof. Using the list of rules 11.1 one checks that a + b is affiliated. Because of 11.11 we know that a^{∗} and b^{∗} are in U. The previous proposition shows that a^{∗} + b^{∗} and therefore (a + b)^{∗} ⊃ a^{∗} + b^{∗} is densely defined. So a + b is closable by 11.1.(ix). Remains to be verified that the closure [a + b] is again affiliated to A. Now the condition ca ⊂ ac for a bounded operator c is equivalent to (c ⊕ c)(Γ_{a}) ⊂ Γ_{a}. Since c ⊕ c is continuous (c ⊕ c)(\overline{Γ_{a}}) ⊂ \overline{Γ_{a}} and it follows that the closure of an affiliated operator is again affiliated. The argument for ab is similar.

So far we have a well-defined addition and multiplication. We still have to check the axioms of an algebra with involution. The crucial point is that a closed extension lying in U is unique. To prove this one first deals with symmetric operators. Again it is important to have a well behaved notion of dimension. Let us first collect a few facts about the Cayley transform of an operator.

Proposition 11.17 (Cayley Transform). Let s be a densely defined symmetric operator. Then the Cayley transform κ(s) is the operator (s − i)(s + i)^{−1} with domain dom(κ(s)) = (s + i)dom(s) and range im(κ(s)) = (s − i)dom(s). This operator is an isometry (on its domain). Moreover κ(s) − 1 is injective and im(κ(s) − 1) = dom(s). If in addition s is closed the subspaces (s ± i)dom(s) are closed. The operator s is selfadjoint if and only if (s + i)dom(s) = H and (s − i)dom(s) = H.

Proof. Compare [73, pages 203-205].

Proposition 11.18. Suppose that s is densely defined, closed, symmetric and affiliated to A. Then s is already selfadjoint.

Proof. First note that all ingredients in the Cayley transform, namely κ(s), dom(κ(s)) and im(κ(s)) are again affiliated to A. From the previous proposition we know that im(κ(s) − 1) = dom(s). Let p = p_{dom(κ(s))} be the projection onto dom(κ(s)) = (s + i)dom(s), then im(κ(s) − 1) = im(κ(s)p − p). From (κ(s)p − p)^{∗} = p^{∗}(κ(s)p − p)^{∗} = p(κ(s)p − p)^{∗} we conclude that im((κ(s)p − p)^{∗}) ⊂ im(p) and therefore

dim((s + i)dom(s)) = dim(dom(κ(s))) = dim(im(p))
≥ dim(im(κ(s)p − p)^{∗}) = dim(im(κ(s)p − p))
= dim(dom(s)) = 1.

Here we use the fact that dom(s) is essentially dense. Since we also know that im((s + i)dom(s)) is a closed subspace it follows that this space is the whole Hilbert space. Since κ(s) is an isometry and affiliated one can also verify that dim(im(κ(s))) = 1. But im(κ(s)) is also closed and therefore it is the whole Hilbert space. This means κ(s) is a unitary operator and therefore s is selfadjoint.

Proposition 11.19 (Unique Closure). Let a ⊂ b be two operators in U, then a = b. In particular, if c is a closable operator whose closure [c] lies in U, then [c] is the only closed extension of c lying in U.

Proof. Let b = us be the polar decomposition of b and set s′ = u^{∗}a ⊂ u^{∗}b = s. Since u^{∗} is bounded s′ is still closed, densely defined and affiliated. It follows that s′ ⊂ s = s^{∗} ⊂ s^{′∗}, i.e. s′ is symmetric. Now the previous proposition implies s′ = s = s^{∗} = s^{′∗} and therefore dom(a) = dom(s′) = dom(s) = dom(b) and a = b. The second assertion follows from the first. Note that [c] is the minimal closed extension of c.

Theorem 11.20. Endowed with these structures U becomes a complex ∗- algebra that contains A as a ∗-subalgebra.

Proof. One has to show that U with this addition multiplication and ∗- operation fulfills the axioms of a ∗-algebra. As an example we will check the distributivity law [a[bc]] = [[ab] + [ac]]. First one verifies similar to 11.15 and 11.16 that ab + ac is densely defined and closable with closure afilliated to A. Now the rules for unbounded operators imply ab + ac ⊂ [ab] + [ac] and ab + ac ⊂ a(b + c) ⊂ a[b + c]. Taking the closure leads to [a[b + c]] ⊃ [ab+ac] ⊂ [[ab]+[ac]]. The above proven uniqueness of a closure in U implies the desired equality.

12 Appendix II: von Neumann Regular Rings

In the first Rings of Operators paper from 1936 [63] Murray and von Neumann studied the lattice of those subspaces of a Hilbert space which are affiliated to a von Neumann algebra (ring of operators). Already in that paper they introduced what we called the algebra of operators affiliated to a finite von Neumann algebra. About the same time von Neumann axiomatized the system of all linear subspaces of a given (projective) space in what he called a continuous geometry [64]. The main new idea was that the notion of dimension (points, lines, planes, ...) need not be postulated but could be proven from the axioms and that dimensions need not be integer valued any more. Of course, the lattice of affiliated subspaces was the main motivating example. He realized that the algebra of affiliated operators gave a completely algebraic description of this lattice in terms of ideals and idempotents. He singled out a class of rings which should play the same role for an abstract continuous geometry and called these rings regular [64] [65] [66]. There is an extensive literature on von Neumann regular rings, but for convenience we will collect some results in this section. More details and further references can be found in [30].

Definition 12.1. A ring R satisfying one of the following equivalent conditions is called von Neumann regular.

(i) For every x ∈ R there exists a y ∈ R, such that xyx = x.

(ii) Every principal right (left) ideal of R is generated by an idempotent.

(iii) Every finitely generated right (left) ideal of R is generated by an idempotent.

(iv) Every finitely generated submodule of a finitely generated projective module is a direct summand.

(v) Every right (left) R-module is flat.

(vi) Tor_{p}^{R}(M, R) = 0 for p ≥ 1 and an arbitrary right R-module M (R has weak- (or Tor-) dimension zero.

(vii) Every finitely presented module is projective.

Proof. Compare [75, Lemma 4.15, Theorem 4.16, Theorem 9.15].

Several processes do not lead out of the class of von Neumann regular rings: Proposition 12.2. (i) If R is von Neumann regular, then the matrix ring

M_{n}(R) is von Neumann regular. (ii) The center of a von Neumann regular ring is von Neumann regular. (iii) A directed union of von Neumann regular rings is von Neumann regu-

lar. Proof. (iii) follows from 12.1(i). For (i) and (ii) see [30, Theorem1.7, Theorem 1.14]. In connection with localization the following properties of von Neumann regular rings are of importance. Proposition 12.3. Let R be a von Neumann regular ring.

(i) Every element a ∈ R is either invertible or a zerodivisor. (ii) A von Neumann regular ring is division closed and rationally closed in

every overring. (iii) If a von Neumann regular ring satisfies a chain condition, i.e. if it is

artinian or noetherian, then it is already semisimple. Proof. (i) and (ii) are treated in 13.15. For (iii) see [30, p.21]. If our ring is a ∗-ring we can make the idempotent in Definition 12.1(ii) unique if we require it to be a projection. Definition 12.4. A von Neumann regular ∗-ring in which a^{∗}a = 0 implies a = 0 is called ∗-regular. The most important property of a ∗-regular ring is the following: Proposition 12.5. In a ∗-regular ring

(i) Every principal right (left) ideal is generated by a unique projection. (ii) Every finitely generated right (left) ideal is generated by a unique pro-

jection. Proof. See [66, Part II, Chapter IV, Theorem 4.5].

Fortunately, in our situation we do not have to care to much about the additional requirement in Definition 12.4.

Proposition 12.6. Let U be the algebra of operators affiliated to a finite von Neumann algebra A. Any subring R of U which is ∗-closed and von Neumann regular is already ∗-regular.

Proof. We only have to ensure that a^{∗}a = 0 implies a = 0. The proof is identical to the one for 2.11.

Another refinement of the notion of von Neumann regularity that is used in Section 3 is unit regularity. One should think of this condition as a finiteness condition, compare [30, Section 4].

Definition 12.7. A ring R is unit regular, if for every x ∈ R there exists a unit u ∈ R such that xux = x.

The following proposition due to Handelman characterizes unit regular rings among all von Neumann regular rings.

Proposition 12.8. Let R be a von Neumann regular ring, then R is unit regular if and only if the following holds: If L, P and Q are finitely generated projective modules, then

P ⊕ L ≅ Q ⊕ L implies P ≅ Q.

Proof. See [34].

In particular for such a ring the map from the semigroup of isomorphism classes of finitely generated projective modules to K_{0}(R) is injective. There seems to be no example of a ∗-regular ring which is not unit regular, compare [30, p.349, Problem 48].

13 Appendix III: Localization of Non-commutative Rings

In this section we will discuss different concepts of localizations for noncommutative rings.

13.1 Ore Localization

Let R be a ring and let X ⊂ R be any subset. A ring homomorphism f : R → S is called X-inverting if f(x) is invertible in S for every x ∈ X.

Definition 13.1. An X-inverting ring homomorphism R → R_{X} is called universal X-inverting if it has the following universal property: Given any X-inverting ring homomorphism f : R → S there exists a unique ring homomorphism ϕ : R_{X} → S such that the following diagram commutes.

R_{X}

[figure: i]
[figure: R; f; S.; ϕ]

As usual this implies that R_{X} is unique up to a canonical isomorphism. One checks existence by writing down a suitable ring using generators and relations. Given a ring homomorphism R → S we denote by T(R → S) the set of all elements in R which become invertible in S. This set is multiplicatively closed. In particular we have the multiplicatively closed set \overline{X} = T(R → R_{X}). Obviously X ⊂ \overline{X} and one verifies that R_{X} and R\overline{X} are naturally isomorphic. Without loss of generality we can therefore restrict ourselves to the case where X is multiplicatively closed, and in that case we usually write T instead of X. Moreover, we will assume that T contains no zerodivisors since this will be sufficient for our purposes and facilitates the discussion. If the ring R is commutative it is well known that there is a model for R_{T} whose elements are fractions, or more precisely, equivalence classes of pairs (a, t) ∈ R × T. If the ring is non-commutative the situation is more complicated. The main difficulty is that once we have decided to work for example with right fractions we have to make sense of a product like

at^{−1}a′t^{′−1}.

Definition 13.2 (Ore Condition). Let T be a multiplicatively closed subset of R. The pair (R, T) satisfies the right Ore condition if given (a, s) ∈ R × T there always exists (b, t) ∈ R × T such that at = sb.

A somewhat sloppy way to remember this is: For every wrong way (left) fraction s^{−1}a there exists a right fraction bt^{−1} with s^{−1}a = bt^{−1}.

Proposition 13.3. Let R be a ring and let T ⊂ R be a multiplicatively closed subset which contains no zerodivisors. Suppose the pair (R, T) satisfies the right Ore condition, then there exists a ring RT^{−1} and a universal T -inverting ring homomorphism i : R → RT^{−1} such that every element of RT^{−1} can be written as i(a)i(t)^{−1} with (a, t) ∈ R × T. The ring RT^{−1} is called a ring of right fractions of R with respect to T.

Proof. Elements in RT^{−1} are equivalence classes of pairs (a, t) ∈ R × T. The pair (a, t) is equivalent to (b, s) if there exist elements u, v ∈ R such that au = bv, su = tv and su = tv ∈ S. For details of the construction and more information see Chapter II in [81].

To avoid confusion we will usually call such a ring an Ore localization. There is of course a left handed version of the above proposition. If both the ring of right and the ring of left fractions with respect to T exist they are naturally isomorphic since both fulfill the universal property. Since we assumed that T contains no zerodivisors the map i is injective and we usually omit it in the notation. We will frequently make use of the following fact.

Note 13.4. Finitely many fractions in RT^{−1} can be brought to a common denominator.

Often one considers the case where T = NZD(R) is the set of all nonzerodivisors in R. In that case, if the ring RT^{−1} exists, it is called the classical ring of fractions (sometimes also the total ring of fractions) of R.

Example 13.5. Let Γ be the free group generated by {x, y}. The group ring ℂΓ does not satisfy the Ore condition with respect to the set NZD(ℂΓ) of all non-zerodivisors: Let ℤ ⊂ Γ be the subgroup generated by x. Now x − 1 is a non-zerodivisor since it becomes invertible in Uℤ and therefore in the overring UΓ of ℂΓ. (In fact every nontrivial element in ℂΓ is a nonzerodivisor since we know from Section 6 that we can embed ℂΓ in a skew field.) The Ore condition would imply the existence of (a, t) ∈ ℂΓ×NZD(ℂΓ) with (y − 1)t = (x − 1)a alias

(x − 1)^{−1}(y − 1) = at^{−1}.

This implies that (−a, t)^{tr} is in the kernel of the map (x − 1, y − 1) : ℂΓ^{2} → ℂΓ. But this map is the nontrivial differential in the cellular chain complex C_{∗}^{cell}(EΓ) where we realize BΓ as the wedge of two circles. Since EΓ is contractible the map must be injective. A contradiction.

For us one of the most important properties of an Ore localization is the following.

Proposition 13.6. Let RT^{−1} be an Ore localization of the ring R, then the functor − ⊗_{R} RT^{−1} is exact, i.e. RT^{−1} is a flat R-module.

Proof. The functor is right exact. On the other hand there is a similar localization functor for modules which is naturally isomorphic to − ⊗_{R} RT^{−1} and which can be verified as being left exact. For more details see page 57 in [81].

We close this section with a statement about the behaviour of an Ore localization under the passage to matrix rings.

Proposition 13.7. Suppose the pair (R, T) satisfies the right Ore condition, then also the pair (M_{n}(R), T · 1_{n}) fulfills the right Ore condition and there is a natural isomorphism

M_{n}(R)(T · 1_{m})^{−1} ≅ M_{n}(RT^{−1}).

Proof. Let ((a_{ij}), t · 1_{n}) ∈ M_{n}(R) × T · 1_{n} be given. Since (R, T) satisfies the Ore condition and finitely many fractions in RT^{−1} can be brought to a common denominator we can find ((b_{ij}), s · 1_{n}) ∈ M_{n}(R) × T · 1_{n} such that in M_{n}(RT^{−1}) we have

(s^{−1} · 1_{n})(a_{ij}) = (s^{−1}a_{ij}) = (b_{ij}t^{−1}) = (b_{ij})t^{−1} · 1_{n}.

We see that (M_{n}(R), T · 1_{n}) fulfills the Ore condition. Now since

M_{n}(R) → M_{n}(RT^{−1})

is T · 1_{n}-inverting the universal property gives us a map

M_{n}(R)(T · 1_{n})^{−1} → M_{n}(RT^{−1}).

Since every matrix in M_{n}(RT^{−1}) can be written as (a_{ij})(t^{−1} · 1_{n}) with (a_{ij}) ∈ M_{n}(R) the map is surjective and injective.

Corollary 13.8. The diagram

[figure: R · 1_{n}; j; RT^{−1} · 1_{n}; M_{n}(R); i; M_{n}(RT^{−1})]

is a push-out in the category of rings and (unit preserving) ring homomorphisms.

Proof. Suppose we are given maps f : M_{n}(R) → S and g : RT^{−1} · 1_{n} → S such that the resulting diagram commutes, then since f ∘i(T ·1_{n}) = g∘j(T ·1_{n}) we see that f is T · 1_{n} inverting. The universal property gives us a unique map

M_{n}(RT^{−1}) ≅ M_{n}(R)(T · 1_{n})^{−1} → S.

One verifies that the resulting diagram commutes.

13.2 Universal Localization

Instead of inverting a subset of the ring one can also invert a given set of matrices or more generally some set of maps between R-modules. This shifts attention from the ring itself to the additive category of its modules or some suitable subcategory. Let Σ be a set of homomorphisms between right R-modules. A ring homomorphism R → S is called Σ-inverting if for every map α ∈ Σ the induced map α ⊗_{R} id_{S} is an isomorphism.

Definition 13.9. A Σ-inverting ring homomorphism i : R → R_{Σ} is called universal Σ-inverting if it has the following universal property: Given any Σ-inverting ring homomorphism R → S there exists a unique ring homomorphism ψ : R_{Σ} → S, such that the following diagram commutes.

R_{Σ}
[figure: i]
[figure: R; f; S.; ψ]

Note that given a ring homomorphism R → S the class Σ(R → S) of all those maps between R-modules which become invertible over S need not be a set. In the following we will therefore always restrict to the category F_{R} of finitely generated free R-modules or the category P_{R} of finitely generated projective R-modules. These categories have small skeleta, and we will always work with those. In the first case we take the set of matrices as such a skeleton where the object set is taken to be the natural numbers. In the second case we take as objects pairs (n, P) with n a natural number and P an idempotent n × n-matrix. If we now look at the set \overline{Σ} = Σ(R → R_{Σ}), the so-called saturation of Σ, we of course have again that R_{Σ} and R\overline{Σ} are naturally isomorphic. The same holds for every set Σ̂ with Σ ⊂ Σ̂ ⊂ \overline{Σ}.

Proposition 13.10. The universal Σ-inverting ring homomorphism exists and is unique up to canonical isomorphism.

Proof. Uniqueness is clear. In the case where Σ is a set of matrices existence can be proven as follows. We start with the free R-ring generated by the set of symbols \{\overline{a_{ij}} | (a_{ij}) = A ∈ Σ\} and divide out the ideal generated by the relations given in matrix form by \overline{A}A = A\overline{A} = 1 where of course \overline{A} = (\overline{a_{ij}}). If Σ is a set of morphisms between finitely generated projectives see Section 4 in [78].

A set of matrices is lower multiplicatively closed if 1 ∈ Σ and a, b ∈ Σ implies

( \begin{matrix}a & 0 \\ c & b \end{matrix} ) ∈ Σ

for arbitrary matrices c of suitable size. We denote by Σ̂ the lower multiplicative closure of a given set Σ. Note that Σ ⊂ Σ̂ ⊂ \overline{Σ} and therefore R_{Σ} ≅ RΣ̂.

Two matrices (or elements) a, b ∈ M(R) are called stably associated over R if there exist invertible matrices c, d ∈ GL(R), such that

c ( \begin{matrix}a & 0 \\ 0 & 1_{n} \end{matrix} ) d^{−1} = ( \begin{matrix}b & 0 \\ 0 & 1_{m} \end{matrix} )

with suitable m and n. There is a similar definition for maps between finitely generated projective modules. The following Proposition has a lot of interesting consequences.

Proposition 13.11 (Cramer’s Rule). Let R be a ring and let Σ be a set of matrices between finitely generated projective R-modules. Every matrix a over R_{Σ} satisfies an equation of the form

s ( \begin{matrix}1 & 0 \\ 0 & a \end{matrix} ) ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) = b

with s ∈ Σ̂, x ∈ M(R_{Σ}) and b ∈ M(R). In particular every matrix over R_{Σ} is stably associated to a matrix over R.

Proof. See [78, Theorem 4.3].

Example 13.12. In the case of a commutative ring the universal localization gives nothing new: Let R be a commutative ring and let Σ be a set of matrices over R. Then R_{Σ} = R_{det(Σ)}, where det(Σ) is the set of determinants of the matrices in Σ, because by Cramer’s rule, which is valid over any commutative ring, a matrix is invertible if and only if its determinant is invertible.

Note 13.13. In general universal localization need not be an exact functor.

13.3 Division Closure and Rational Closure

Given a ring S and a subring R ⊂ S we denote by T(R ⊂ S) the set of all elements in R which become invertible over S and by Σ(R ⊂ S) the set of all matrices which become invertible over S. Both sets are multiplicatively closed. One can consider abstractly the universal localizations R_{T(R⊂S)} and R_{Σ(R⊂S)}. The question arises whether these rings can be embedded in S. Unfortunately this is not always possible. The intermediate rings in the following definition serve as potential candidates for embedded versions of R_{T(R⊂S)} and R_{Σ(R⊂S)}.

Definition 13.14. Let S be a ring.

(i) A subring R ⊂ S is called division closed in S if

T(R ⊂ S) = R^{×},

i.e. for every element r ∈ R which is invertible in S the inverse r^{−1} already lies in R.

(ii) A subring R ⊂ S is called rationally closed in S if

Σ(R ⊂ S) = GL(R),

i.e. for every matrix A over R which is invertible over S the entries of the inverse matrix A^{−1} are all in R.

(iii) Given a subring R ⊂ S the division closure of R in S denoted by D(R ⊂ S) is the smallest division closed subring containing R.

(iv) Given a subring R ⊂ S the rational closure of R in S denoted by R(R ⊂ S) is the smallest rationally closed subring containing R.

Note that the intersection of division closed intermediate rings is again division closed, and similarly for rationally closed rings. In [15, Chapter 7, Theorem 1.2] it is shown that the set

{a_{i,j} ∈ S | (a_{i,j}) invertible over S, (a_{i,j})^{−1} matrix over R }

is a subring of S and that it is rationally closed. Since this ring is contained in R(R ⊂ S) the two rings coincide. From the definitions we see immediately that the division closure is contained in the rational closure. The next proposition is very useful if one has to decide whether a given ring is the division closure respectively the rational closure.

Proposition 13.15. A von Neumann regular ring R is division closed and rationally closed in every overring.

Proof. Suppose a ∈ R is not invertible in R, then the corresponding right R-linear map l_{a} : R → R is not an isomorphism. Therefore the kernel or the cokernel is nontrivial. Both split off as direct summands because the ring is von Neumann regular. The corresponding projection on the kernel or cokernel is given by left-multiplication with a suitable idempotent. This idempotent shows that a must be a zerodivisor. A zerodivisor cannot become invertible in any overring. We see that R is division closed. A matrix ring over a von Neumann regular ring is again von Neumann regular. Hence the same reasoning applied to the matrix rings over R yields that R is rationally closed in every overring. Since the division closure is contained in the rational closure the last statement follows.

Note 13.16. In particular we see that once we know that the division closure D(R ⊂ S) is von Neumann regular it coincides with the rational closure R(R ⊂ S).

The following proposition relates the division closure respectively the rational closure to the universal localizations R_{T(R⊂S)} and R_{Σ(R⊂S)}.

Proposition 13.17. Let R ⊂ S be a ring extension.

(i) The map ϕ : R_{T(R⊂S)} → S given by the universal property factorizes over the division closure.

R_{T(R⊂S)}

[figure]

R^{⊂} D(R ⊂ S) Φ_{}^{⊂}^{}S

(ii) If the pair (R, T(R ⊂ S)) satisfies the right Ore condition, then Φ is an isomorphism.

(iii) The map ψ : R_{Σ(R⊂S)} → S given by the universal property factorizes over the rational closure.

R_{Σ(R⊂S)}

[figure]

[figure: R; ⊂; R(R ⊂ S); Ψ; ⊂; S]

The map Ψ is always surjective.

Proof. (i) All elements of T(R ⊂ S) are invertible in D(R ⊂ S) by definition of the division closure. Now apply the universal property. (ii) Note that T(R ⊂ S) always consists of non-zerodivisors. Thus we can choose a ring of right fractions as a model for R_{T(R⊂S)}. Every element in imϕ is of the form at^{−1}. Such an element is invertible in S if and only if a ∈ T(R ⊂ S). We see that imϕ is division closed and Φ is surjective. On the other hand the abstract fraction at^{−1} ∈ RT(R ⊂ S)^{−1} is zero if and only if a = 0 because T(R ⊂ S) contains no zero divisors, so Φ is injective. (iii) That the map ψ factorizes over R(R ⊂ S) follows again immediately from the universal property. By Cohn’s description of the rational closure R(R ⊂ S) we need to find a preimage for a_{i,j} where (a_{i,j}) is a matrix invertible over S whose inverse lives over R. The generator and relation construction of the universal localization given above immediately yields such an element.

In general it is not true that the map Ψ is injective. In 13.21 we give a counterexample where S is even a skew field.

13.4 Universal R-Fields

We describe in this subsection a very special case of ring extensions, where the rational closure R(R ⊂ S) is indeed isomorphic to the universal localization R_{Σ(R⊂S)}. Again the main ideas are due to Cohn [15]. Given a commutative ring R without zerodivisors there is always a unique field of fractions. It is characterized by the fact that it is a field which contains R as a subring and is generated as a field by R. One possibility for a non-commutative analogue is the following definition. Note that following Cohn we use the terms field, skew field and division ring interchangeably.

Definition 13.18. Let R be an arbitrary ring. A ring homomorphism R → K with K a skew field is called an R-field. It is called an epic R-field if K is generated as a skew field by the image of R. An epic R-field is called a ring of fractions for R if the homomorphism is injective.

The problem with this definition is that a ring of fractions is not unique. In 13.21 we will see that the non-commutative polynomial ring ℂ < x, u > admits several non isomorphic rings of fractions. Let us again look at the commutative case. The ring R = ℤ admits other epic R-fields than ℚ, namely the residue fields ℤ → 𝔽_{p}. The diagram

R_{Σ(R→K)}

[figure]

[figure: R; K]

becomes in this special case

ℤ_{(p)}

[figure]

[figure: ℤ; 𝔽_{p},]

where ℤ_{(p)} is the local ring ℤT^{−1} with T = ℤ − (p). One interprets the map ℤ_{(p)} → 𝔽_{p} as a morphism from the ℤ-field ℚ to the ℤ-field 𝔽_{p} in a suitable category, a so called specialization. There is a unique specialization from the ℤ-field ℚ to every other epic ℤ-field. The next definition takes these observations as a starting point.

Definition 13.19. A local homomorphism from the R-field K to the R-field L is an R-ring homomorphism f : K_{0} → L where K_{0} ⊂ K is an Rsubring of K and K_{0} −ker f = K_{0}^{×}. The ring K_{0} is a local ring with maximal ideal ker f. Two local homomorphisms from K to L are equivalent if they restrict to a common homomorphism which is again local. An equivalence class of local homomorphisms is called a specialization. An initial object in the category of epic R-fields and specializations is called a universal Rfield. If moreover the map R → K is injective it is called a universal field of fractions.

A universal R-field is unique up to isomorphism, but even if there exists a field of fractions for R there need not exist a universal R-field ([15, p.395, Exercise 10]). On the other hand, if a universal R-field exists the map need not be injective, i.e. it need not be a universal field of fractions. Now given a ring of fractions for R one can ask whether in the diagram

R_{Σ(R⊂K)}

[figure]

[figure: R; ψ; K; Ψ]

Ψ is an isomorphism. Note that if we have a ring extension R ⊂ K with K a skew field, then D(R ⊂ K) is the subfield generated by R in K, and from Proposition 13.15 we know that D(R ⊂ K) = R(R ⊂ K). In particular, since a field of fractions is an epic R-field we have K = D(R ⊂ K) = R(R ⊂ K). As we already know from the discussion in the last subsection Ψ is surjective. One might hope that Ψ is an isomorphism:

(i) For every field of fractions R → K.

(ii) At least for universal fields of fractions R → K.

Unfortunately there are counterexamples in both cases. It is claimed in [4, p.332] that there exists a counterexample to (ii). We give a concrete counterexample to (i) in 13.21. After all these bad news about the noncommutative world now a positive result due to Cohn. A ring is a semifir if it has invariant basis number and every finitely generated submodule of a free module is free.

Proposition 13.20. Let R be a semifir, then there exists a universal field of fractions and the map R_{Σ(R→K)} → K is an isomorphism.

Proof. By definition of a semifir every finitely generated projective module is free and has a well-defined rank. A map R^{n} → R^{n} between finitely generated free modules is called full if it does not factorize over a module of smaller rank. Now [15, Chapter 7, 5.11] says that there is a universal field of fractions R → K such that every full matrix (alias map between finitely generated projectives) becomes invertible over K. Since non full matrices can not become invertible over K we see that the set of full matrices over R coincides with Σ(R → K). Now [15, Chapter 7, Proposition 5.7 (ii)] yields that the map R_{Σ(R→K)} → K is injective.

The main examples of semifirs are non-commutative polynomial rings over a commutative field and the group ring over a field of a finitely generated free group [16, Section 10.9]. These rings are even firs. Before we go on, here the promised counterexample. Compare [4]. Example 13.21. Let ℂ < x, y, z > and ℂ < x, u > be non-commutative polynomial rings. The map defined by x ↦ x, y ↦ xu and z ↦ xu^{2} is injective. Since ℂ < x, u > is a semifir there is a universal field of fractions ℂ < x, u >→ K. Now the division closure D of the image of ℂ < x, y, z > in K is a field of fractions for ℂ < x, y, z >. Consider the diagram

ℂ < x, y, z >_{Σ(ℂ<x,y,z>⊂D)}

[figure]

[figure: ℂ < x, y, z >; ψ; D; Ψ]

given by the universal property. The map Ψ is not injective since, for example, the element y^{−1}z − x^{−1}y is mapped to zero. From Proposition 13.20 we know that for the universal field of fractions of ℂ < x, y, z > the corresponding map is an isomorphism. So D is not the universal field of fractions for ℂ < x, y, z >, and in particular we have found two non-isomorphic fields of fractions for ℂ < x, y, z >.

13.5 Recognizing Universal Fields of Fractions

Let Γ be the free group on two generators. From the preceding subsection we know that the group ring ℂΓ, which is a semifir, admits a universal field of fractions. The following gives a criterion to decide whether a given field of fractions for ℂΓ, as for example D(ℂΓ ⊂ UΓ), is the universal field of fractions. First we need some more notation. Let Γ be a group and let G be a finitely generated subgroup. We say that G is indexed at t ∈ G if there exists a homomorphism p_{t} : G → ℤ which maps t to a generator. In that situation we have an exact sequence

1 → ker p_{t} → G → ℤ → 1

together with a homomorphic section given by 1 ↦ t. Therefore the group ring RG ⊂ RΓ has the structure of a skew polynomial ring.

RG = (R ker p_{t}) ∗ ℤ.

Definition 13.22. Let Γ be a free group. An embedding ℂΓ ⊂ D into a skew field (e.g. a ℂΓ-field of fractions) is called Hughes-free if for every finitely generated subgroup G and every t ∈ G at which G is indexed we have that the set {t^{i} | i ∈ ℤ} is D(ℂ ker p_{t} ⊂ D)-left linearly independent.

A particular case of the main theorem in [37] states:

Proposition 13.23. Any two ℂΓ-fields of fractions which are Hughes-free are isomorphic as ℂΓ-fields.

This is useful since Lewin shows in [45] the following proposition.

Proposition 13.24. The universal field of fractions of the group ring ℂΓ of the free group on two generators is Hughes-free.

Combining these results with Proposition 13.20 we get:

Corollary 13.25. Let Γ be a free group. If ℂΓ ⊂ D is a Hughes-free ℂΓfield of fractions, then it is a universal ℂΓ-field of fractions, and it is also a universal localization with respect to Σ(ℂΓ ⊂ D).

13.6 Localization with Respect to Projective Rank Functions

So far we were mostly interested in universal localizations R_{Σ(R⊂S)} where a ring extension R ⊂ S was given. But there are other ways to describe reasonable sets Σ one would like to invert. We are following Schofield [78].

Definition 13.26. Let R be a ring. A projective rank function for R is a homomorphism ρ : K_{0}(R) → ℝ such that for every finitely generated projective R-module P we have ρ([P]) ≥ 0 and ρ([R]) = 1. The rank function is called faithful if ρ([P]) = 0 implies P = 0.

Note that such a rank function is in general not monotone, i.e. P ⊂ Q does not imply ρ(P) ≤ ρ(Q) as it is the case for the dimensionfunction discussed in Section 3. Given a projective rank function we define the inner rank of a map α : P → Q between finitely generated projectives as

ρ(α) = inf{ρ(P′) | α factorizes over the f.g. projective module P′}.

Definition 13.27. A map α : P → Q is called right full if ρ(α) = ρ(Q) and left full if ρ(α) = ρ(P). It is called full if it is left and right full.

Note 13.28. If R is semihereditary and the rank function is monotone, then ρ(α) = ρ([imα]). If R is von Neumann regular, then every rank function is monotone and could be called a dimension function if it is also faithful. In that case the notions right full, left full and full correspond to surjective, injective and bijective.

Again it is the free group which shows that in general these notions give something new.

Example 13.29. Let Γ be the free group on two generators x and y. Take ρ as the composition

ρ : K_{0}(ℂΓ) → K_{0}(UΓ) →^{dim} ℝ.

Now look at the map

ℂΓ^{2} ⟶^{(x−1,y−1)} ℂΓ.

This is the differential in the cellular chain complex of the universal covering of BΓ = S^{1} ∨ S^{1}. This map is right full, but not surjective, and it is injective, but not left full with respect to ρ.

It is reasonable to invert universally all matrices (or maps between finitely generated projective modules) which are full with respect to a given projective rank function ρ. The following lemma tells us that in one of our standard situations we get nothing new.

Lemma 13.30. Let R → S be a ring homomorphism with S von Neumann regular and suppose there is a faithful projective rank function dim : K_{0}(S) → ℝ for S. Let

ρ : K_{0}(R) → K_{0}(S) → ℝ

be the induced projective rank function for R. Then a map α : P → Q which becomes an isomorphism over S is a full map.

Proof. Suppose α ⊗_{R} id_{S} is an isomorphism. If α factorizes over the finitely generated projective module P′, then in the induced factorization after tensoring we have

ρ(α) ≤ ρ(P) = dim(P ⊗_{R} S) ≤ dim(P′ ⊗_{R} S).

Taking the infimum over all P′ gives ρ(α) = ρ(P) = ρ(Q). The map is full.

The following theorem of Schofield is very useful for our purposes:

Theorem 13.31. Let R be semihereditary and let ρ : K_{0}(R) → ℝ be a faithful projective rank function such that every map between finitely generated projectives factors as a right full followed by a left full map. If Σ is a collection of maps that are full with respect to the rank function, then K_{0}(R) → K_{0}(R_{Σ}) is surjective.

Proof. See [78, Theorem 5.2]. Note that over a semihereditary ring all rank functions are automatically Sylvester projective rank functions by [78, Lemma 1.1, Theorem 1.11].

The interesting example of this situation is not the extension N Γ ⊂ UΓ, but the extension ℂΓ ⊂ UΓ in the case where Γ is a free group or a finite extension of a free group. Over a hereditary ring the condition about the factorization is always fulfilled ([78, Corollary 1.17]). In general the following observation sheds some light on this condition.

Note 13.32. Let R be semihereditary and ρ : K_{0}(R) → ℝ be a projective rank function with im(ρ) ⊂ \frac{1}{l}ℤ for some integer l. Every map between finitely generated projectives factors as a right full followed by a left full map.

Proof. Let α : P → Q be a map between finitely generated projectives. Over a semihereditary ring ρ(α) = inf{ρ(P′) | imα ⊂ P′ ⊂ P, P′ f.g. projective} since the map from an abstract P′ to its image is always split. So there must be a finitely generated projective submodule P′ with im(α) ⊂ P′ ⊂ Q such that ρ(P′) = ρ(α) = inf{ρ(P′)|im(α) ⊂ P′ ⊂ Q}. In the factorization P → P′ → Q the first map is right full and the second is left full.

Let us summarize what we can achieve in this abstract set-up.

Corollary 13.33. Let R ⊂ S be a semihereditary subring of the von Neumann regular ring S. Suppose there is a faithful dimension function K_{0}(S) → \frac{1}{l}ℤ for some integer l. Then the diagram

K_{0}(R_{Σ(R⊂S)})
[figure: i_{∗}]
[figure: K_{0}(R); K_{0}(R(R ⊂ S)); Ψ_{∗}; K_{0}(S); →; \frac{1}{l}ℤ ⊂ ℝ]

is commutative and the map i_{∗} is surjective.

Unfortunately it is not clear whether or not Ψ_{∗} is surjective. 134

14 Appendix IV: Further Concepts from the Theory of Rings

14.1 Crossed Products

If H is a normal subgroup of the group Γ, the group ring ℂΓ can be described in terms of the ring ℂH and the quotient G = Γ/H. This situation is axiomatized in the notion of a crossed product. For a first understanding of the following definition one should consider the case where the ring R is ℂG, the ring S is ℂΓ and the map μ is a set theoretical section of the quotient map Γ → G = Γ/H followed by the inclusion Γ ⊂ ℂΓ.

Definition 14.1 (Crossed Product). A crossed product R ∗ G = (S, μ) of the ring R with the group G consists of a ring S which contains R as a subring together with an injective map μ : G → S^{×} such that the following holds.

(i) The ring S is a free R-module with basis μ(G).

(ii) For every g ∈ G conjugation map

c_{μ(g)} : S → S, s ↦ μ(g)sμ(g)^{−1}

can be restricted to R.

(iii) For all g, g′ ∈ G the element

τ(g, g′) = μ(g)μ(g′)μ(gg′)^{−1}

lies in R^{×}.

In this definition we consider a crossed product as an additional structure on the given ring S = R ∗ G. Alternatively one can consider a crossed product as a new ring constructed out of a given ring R, a group G and some twisting data. This will be the content of Proposition 14.2 below. The first condition in the above definition tells us that every element of R∗G can be written in the form

∑_{g∈G} r_{g}μ(g), with r_{g} ∈ R,

where only finitely many of the r_{g} are nonzero. The multiplication is determined by

rμ(g) · r′μ(g′) = rc_{μ(g)}(r′)τ(g, g′)μ(gg′).

By associativity [rμ(g) · r′μ(g′)]·r′′μ(g′′) = rμ(g)·[r′μ(g′) · r′′μ(g′′)]. Writing this out yields the following equation in R

rc_{μ(g)}(r′) [(c_{τ(g,g′)} ∘ c_{μ(gg′)})(r′′)] τ(g, g′)τ(gg′, g′′) =
rc_{μ(g)}(r′) [(c_{μ(g)} ∘ c_{μ(g′)})(r′′)] c_{μ(g)}(τ(g′, g′′))τ(g, g′g′′).

Setting r = r′ = r′′ = 1 yields

τ(g, g′) · τ(gg′, g′′) = c_{μ(g)}(τ(g′, g′′)) · τ(g, g′g′′).

Setting r = r′ = 1 and cancelling the τ’s yields

c_{τ(g,g′)} ∘ c_{μ(gg′)} = c_{μ(g)} ∘ c_{μ(g′)}.

On the other hand these two equations imply the above. This leads to the following proposition.

Proposition 14.2 (Construction of a Crossed Product). Given a ring R, a group G and maps

\overline{c} : G → Aut(R), g ↦ \overline{c_{μ(g)}}, \overline{τ} : G × G → R^{×},

such that

\overline{τ}(g, g′) · \overline{τ}(gg′, g′′) = \overline{c_{μ(g)}}(\overline{τ}(g′, g′′)) · \overline{τ}(g, g′g′′)
c_{\overline{τ}(g,g′)} ∘ \overline{c_{μ(gg′)}} = \overline{c_{μ(g)}} ∘ \overline{c_{μ(g′)}}

holds, we can construct a ring R∗_{\overline{τ},\overline{c}}G together with a map μ : G → R∗_{\overline{τ},\overline{c}}G such that μ is a crossed product structure on R∗_{\overline{τ},\overline{c}}G and c_{μ(g)} = \overline{c_{μ(g)}} and τ = \overline{τ}.

Proof. Take the free R-left module on symbols μ(g) and define a multiplication as above. The conditions make sure that this multiplication is associative.

Usually one omits τ, c and μ in the notation. According to the situation we will write R ∗ G, (R ∗ G, μ) or R∗_{τ,c}G. The following proposition describes homomorphisms out of a crossed product.

Proposition 14.3. Let R∗_{τ,c}G be a crossed product and let S be a ring. Suppose we are given a ring homomorphism f : R → S and a map ν : G → S^{×} such that

[figure: G × G; τ; R; R; S]
[figure: τ_{ν}]  and
[figure: S; R; c_{μ(g)}; S; c_{ν(g)}]

commute. Then there exists a unique ring homomorphism

F : R∗_{τ,c}G → S

such that F extends f and F ∘μ = ν. Conversely a homomorphism R∗_{τ,c}G → S determines f and ν (and therefore τ_{ν} and c_{ν}).

In particular if we apply this to the case where S is a crossed product we get the notion of a crossed product homomorphism. Again we can either consider such a crossed product homomorphism f ∗ G : R ∗ G → R̃ ∗ G as a usual ring homomorphism with additional properties or as a new ring homomorphism built out of f : R → R̃.

Proposition 14.4 (Crossed Product Homomorphism). Let (R ∗ G, μ) and (R̃ ∗G, μ̃) be crossed products. A ring homomorphism F : R ∗G → R̃ ∗G is called a crossed product homomorphism if

(i) The following diagram commutes

R ∗ G
[figure: μ]
[figure: G; μ̃; R̃ ∗ G.; F]

(ii) The map F restricts to a map R → R̃.

On the other hand, given a ring homomorphism f : R → R̃ such that

[figure: G × G; τ; R; R; R̃]
[figure: τ̃]  and
[figure: R̃; f; R; c_{μ(g)}; R̃; c̃_{ν̃(g)}]

commute, there exists a unique crossed product homomorphism

f ∗ G : R∗_{τ,c}G → R̃∗_{τ̃,c̃}G

which extends f.

The following propositions in the main text also deal with crossed products: Proposition 8.5 on page 67 (Crossed products and localizations), Proposition 8.12 on page 75 (Crossed products and chain conditions) and Lemma 9.4 on page 95.

14.2 Miscellaneous

In this subsection we collect some definitions from ring theory which are used throughout the text.

Definition 14.5. Let R be a ring and f : R^{n} → R^{m} be any R-linear map between finitely generated free right R-modules. The following can be taken as a definition of the notions coherent, semihereditary, semifir, von Neumann regular.

(i) R is coherent, if im(f) is always finitely presented.

(ii) R is semihereditary, if im(f) is always projective.

(iii) R is a semifir, if im(f) is always a free module and the ring has invariant basis number.

(iv) R is von Neumann regular, if im(f) is always a direct summand of R^{m}.

A ring R has invariant basis number if every finitely generated R-module has a well-defined rank, i.e. R^{n} ≅ R^{m} implies n = m.

Definition 14.6. (i) A ring R is called prime if there are no zerodivisors on the level of ideals, i.e. if I and J are two sided ideals then IJ = 0 implies I = 0 or J = 0.

(ii) A ring R is called semiprime if there are no nilpotents on the level of ideals, i.e. if I is a two sided (or one sided) ideal, then I^{n} = 0 implies I = 0. It is sufficient to check the case n = 2.

A semiprime ring is a subdirect product of prime rings [76, p.164]. The remarks on page 186 in [76] might help to relate semiprimeness to other ring theoretical concepts. For the notions of artinian, noetherian and semisimple rings we recommend pages 495–497 in [76] where one finds a list with characterizations and basic properties of these notions.

Glossary of Notations

(A) and (B) see 5.10 and 8.3
(C)  see 8.4

Groups

Γ  ’generic’ group
G  ’generic’ subgroup of Γ
H  ’generic’ quotient of Γ
g,h or k  ’generic’ element of a group
e  unit of a group
G ∗ H  free product of groups
Aut(G)  automorphism group of G
\frac{1}{|FinΓ|}ℤ  see page 31
X or Y  ’generic’ class of groups

EG class of elementary amenable

groups, see Definition 7.1

C Linnell’s class of groups, see Def-

inition 7.5

C′ groups in C with a bound on the

order of finite subgroups

N F class of groups not containing ℤ ∗ ℤ

Rings

R or S  ’generic’ ring
R ⊂ S  ’generic’ ring extension
R^{×}  multiplicative group of units
M_{n}(R)  ring of n × n matrices over R

GL_{n}(R) group of invertible n × n-matrices

over R

e or f  ’generic’ idempotent
p or q  ’generic’ projection in a ∗-ring

T ’generic’ multiplicatively closed

subset of R

NZD(R)  set of all non-zerodivisors in R

T(R → S) see remarks after Definition 13.1

T(R ⊂ S) special case with R → S embed-

ding of a subring

Σ  ’generic’ set of matrices over R
Σ(R → S) see remarks after Definition 13.9

Σ(R ⊂ S) special case with R → S embed-

ding of a subring

R_{X} universal X-inverting ring, see

Definition 13.1

R_{T}  see remarks after Definition 13.1

R_{Σ} universal Σ-inverting ring, see

Definition 13.9

D(R ⊂ S) division closure of R in S, see Def-

inition 13.14

R(R ⊂ S) ratioonal closure of R in S, see

Definition 13.14

R ∗ G crossed product of R with G, see

Definition 14.1

μ crossed product structure map,

see Definition 14.1

c_{r}  conjugation map x ↦ rxr^{−1}
τ(g, g′)  see Definition 14.1

f ∗ G crossed product homomorphism, see 14.4

Specific rings

ℂΓ group ring with complex coeffi-

cients

RΓ group ring with R-coefficients, see

Conjecture 5.3

A ’generic’ von Neumann algebra,

see page 6

N Γ group von Neumann algebra, see

page 2

U algebra of operators affiliated to a A, see Definition 11.10 UΓ algebra of operators affiliated to

N Γ, see 2.1

SΓ  see Theorem 5.10
T(Γ)  see page 62
Σ(Γ)  see page 62
D(Γ)  see page 62
R(Γ)  see page 62
B(H)  algebra of bounded operators
M′  commutant, see page 6
M′′  double commutant
W^{∗}(s)  see page 110

Ł^{∞}(X, μ) algebra of essentially bounded

functions

Ł(X, μ) = Ł(X) algebra of measurable functions, see Example 2.3

Modules

M  ’generic’ module

P or Q ’generic’ finitely generated mod-

ule

tM torsion submodule, see Defini-

tion 3.14

TM largest zero-dimensional submod-

ule, see Definition 3.15

PM  see Definition 3.15

Homology

H_{p}^{Γ}(X; N Γ)  see page 29
H_{p}^{Γ}(X; UΓ)  see Definition 4.1
b_{p}^{(2)}(Γ)  L^{2}-Betti numbers, see page 29
χ(Γ)  Euler characteristic of BΓ

χ_{virt}(Γ) virtual Euler characteristic, see

page 34

χ^{(2)}(Γ) L^{2}-Euler characteristic, see

page 34

Tor_{p}^{R}(−; M)  derived functors of − ⊗_{R} M

K-Theory

K_{0}(R) Grothendieck group of isomorphism classes of f.g. proj. Rmodules

G_{0}(R)  see page 83
K_{1}(R)  see Definition 10.3
K_{1}^{inj}(R)  see Definition 10.3
K_{1}^{w}(A)  see Definition 10.3
colim_{K∈FinΓ}K_{0}(ℂK) see 5.3

Lattices

L  ’generic’ lattice, see page 12

L_{P roj}(A) lattice of projections in a von

Neumann algebra

L_{Hilb}(M) lattice of closed Hilbert A-

submodules of M

L_{all}(M_{R})  lattice of all submodules of M

L_{fg}(P_{R}) set of all finitely generated sub-

modules of P_{R}

L_{ds}(P_{R}) set of all submodules of P_{R} which are direct summands

Miscellaneous

E(Γ, Fin)  see page 39
HC_{0}(R)  see proof of Proposition 5.19
H  ’generic’ Hilbert space
< x, y >  scalar product on a Hilbert space

l^{2}X Hilbert space with orthonormal

basis X

l^{2}(A)  see page 13

Ł^{2}(X; μ) Hilbert space of square integrable

functions

L^{0}(H) ideal of finite rank operators in

B(H)

L^{1}(H)  ideal of trace class operators
L^{p}(H)  Schatten ideals, see page 45

tr(a)  trace of a trace class operator
ρ, ρ_{±}  see Definition 6.1
A^{p}  see Proposition 6.3
N Γ^{0}  see Proposition 6.3
N Γ^{p}  see Proposition 6.3
tr_{A}

trace of a finite von Neumann algebra, see page 6

tr_{Z(A)}  center valued trace, see page 98
tr_{M_{n}(A)}  see page 18
dim_{U}

see Proposition 3.9 and Theorem 3.12

dim′_{U}  see Definition 3.10
det_{F K}

Fuglede-Kadison determinant, see page 102

T_{R→R_{Σ}}  see Definition 10.7

p ∼_{MvN} q Murray von Neumann equivalence of projections, see Lemma 3.6

dom(a) domain of an operator
a ⊂ b  b is an extension of a, see page 106

spec(a) spectrum of an operator, see

page 107

κ(s) Cayley transform of s, see Propo-

sition 11.17

Proj(R) monoid of isomorphism classes of f.g. proj. R-modules

Index

AW^{∗}-algebra, 14

K_{1} of, 84 G-theory, 69 G_{0}

and K_{0}, 69 of an Ore localization, 69

K-theory, 81 and localization, 85 negative, 34

K_{0} and G_{0}, 69 of algebra of affiliated opera-

tors, 81 of finitely presented torsion mod-

ules, 86 of unit regular ring, 97 of von Neumann algebra, 81

K_{1} of AW^{∗}-algebra, 84 of algebra of affiliated opera-

tors, 84 of injective endomorphisms, 83 of unit regular ring, 84 of von Neumann algebra, 84

L^{2}-Euler characteristic, 29 L^{2}-homology, 24

universal coefficient theorem for,

77 R-field, 105

epic, 105 X-inverting ring homomorphism, 98 Γ-space, 24

with finite isotropy, 77 Σ-inverting ring homomorphism, 101 ∗-closed

subrings of UΓ, 62 ∗-regular ring, 9

definition of, 96

abelianized group of units, 84 affiliated

operator, 89 subspace, 89

affiliated operator

characterization of, 90 algebra

AW^{∗}, 14 matrix, 14 von Neumann, 5

algebra of affiliated operators K_{0} of, 81 K_{1} of, 84 definition of, 91

almost equivariant, 40 amenable

group, 44 von Neumann algebra, 45

amenable group characterization of, 45 Folner criterion, 45

arbitrary module, 21

dimension for, 17 artinian ring, 62, 114

and crossed product, 62 assembly map, 33 Atiyah conjecture, 26

and directed unions, 27 and division closure, 32 and subgroups, 28 for the free group on two gen-

erators, 42 for torsionfree groups, 32 iff DΓ skew field, 74 strategy for, 29 with coefficients, 27

Atiyah-Hirzebruch spectral sequence, 33

Bass trace map, 35 Bredon homology, 33

category

of finitely generated free mod-

ules, 101 of finitely generated projective

modules, 101 of finitely presented torsion mod-

ules, 85 orbit, 33

Cayley graph, 39, 45 transform, 93

center

of von Neumann regular ring,

95 valued dimension, 82 valued trace, 81

chain condition

and von Neumann regular rings,

96 Chern character, 35 classes of groups, 43 classical ring of fractions

definition of, 99 of ℂΓ, 53

classifying space for a family, 33 closed

extension of an operator, 87 operator, 87

closure division, 32, 52, 103 rational, 32, 52, 103

cofinal-measurable, 22 coherent ring, 11, 20

definition of, 114 common complement, 13 commutant, 5, 89 commutative ring

field of fractions of, 104 commute, 89 complement

common, 13 complete lattice, 10, 11, 16 composition

of unbounded operators, 87 condition

Ore, 8 conjecture

Atiyah, 26 Atiyah with coefficients, 27 isomorphism, 33 Kadison, 37 Serre’s, 28 zero divisor, 28

construction

GNS, 10 core of an operator, 88 counterexample

AG ≠ N F, 45 EG ≠ AG, 45 SΓ not semisimple, 31 SΓ not unique, 32 field of fractions not universal,

106 injective but not left full, 108 non-isomorphic fields of fractions,

106 nontrivial module with vanish-

ing dimension, 22 right full but not surjective, 108 to Ψ injective, 106 to Ψ injective for universal field

of fractions, 106 to Ore condition, 99

Cramer’s rule, 60, 85, 102 crossed product

and artinian rings, 62 and localization, 56 and noetherian rings, 62 and polycyclic-by-finite groups,

62 and semiprime rings, 62 construction of, 112 definition of, 111 homomorphism, 113 semisimple, 63

dense

essentially, 91 determinant

Fuglede-Kadison, 84 dimension, 15

additivity of, 17 center valued, 82 cofinality of, 17 of affiliated subspace, 91 of arbitrary U-module, 17 of finitely generated projective

A-module, 15 of finitely generated projective

U-module, 15 of Hilbert A-module, 15 projective, 85 Tor-, 6 weak-, 6 directed union, 4, 71

and Atiyah conjecture, 27 division closed

definition of, 102 division closure, 32, 52

and Atiyah conjecture, 32 definition of, 103

division ring, 105

domain

of an operator, 87 double commutant theorem, 5

elementary amenable groups, 43 and Tor_{p}^{ℂΓ}(−; DΓ), 77 and Ore localization, 53

elements

stably associated, 60 epic R-field, 105 equivalent

Murray von Neumann, 13 equivariant

almost, 40 essential

homomorphism, 89 essentially bounded function, 3, 6 essentially dense, 91 Euler characteristic, 28

L^{2}-, 29

for groups in C, 78

virtual, 28 exact functor, 8, 99 example

Ψ not injective, 106 SΓ not semisimple, 31 SΓ not unique, 32 field of fraction not universal,

106 for AG ≠ N F, 45 for EG ≠ AG, 45 group in C, 46 infinite cyclic group, 3 injective but not left full, 108 non-isomorphic fields of fractions,

106 nontrivial module with vanish-

ing dimension, 22 not satisfying Ore condition, 99 of semifirs, 106 right full but not surjective, 108 universal localization of com-

mutative ring, 102 extension

and localization, 58 of an operator, 87

faithful trace, 5, 15 field, 105 field of fractions

of a commutative ring, 104 universal, 105

finite homological type, 28 finite trace, 5 finitely generated projective mod-

ule, 20 finitely presented module, 20 finitely presented torsion modules,

85 K_{0} of, 86

flat, 8, 30, 45, 99 fraction

of operators, 7 Fredholm module, 37, 38

p-summable, 38

construction of, 40 free

Hughes-, 107 free group, 74

and DΓ, 74 finite extensions of, 75

free group on two generators, 37,

99 Atiyah conjecture for, 42 geometric properties of, 39 not amenable, 44

free product, 49 Fuglede-Kadison determinant, 84 full, 108 function

essentially bounded, 3, 6 measurable, 6 square integrable, 6 step, 82

function calculus, 89 functor

exact, 8, 99 left exact, 20

Fölner criterion, 45

GNS construction, 10 Goldie’s theorem, 64 graph

Cayley, 39 of an operator, 87

group amenable, 44 elementary amenable, 43 free, 74 free on two generators, 37 infinite cyclic, 3 infinite locally finite, 31 of units, 84 torsionfree, 26, 31 virtually abelian, 81 virtually cyclic, 36

group von Neumann algebra, 1 of infinite cyclic group, 3 operators affiliated to, 90 type of, 81

groups X -by-Y, 43, 53 classes of, 43 locally-X -, 43, 53

Hattori-Stallings rank, 35 hereditary ring, 76, 109 Hilbert A-module, 10 Hilbert A-modules

lattice of, 11 homological type

finite, 28 homology

L^{2}-, 24 Bredon, 33 generalized equivariant, 33 reduced L^{2}-, 24 unreduced L^{2}-, 24 with twisted coefficients, 24

homomorphism crossed product, 113 essential, 89 local, 105

Hughes-free, 107

idempotent, 4, 12, 28 induction

set-up, 61 transfinite, 43, 46

induction map, 29 induction principle

for C, 47
for EG, 44

induction theorem

Moody’s, 36, 70 infinite cyclic group, 3 infinite locally finite group, 31 invariant basis number, 114 invariant mean, 44, 45 invertible operator, 7 isomorphism

lattice, 10, 11 weak, 7, 83

isomorphism classes of projections,

15 isomorphism conjecture, 33, 76

consequences of, 34 isomorphism of projections, 13 iterated localization, 59

K-theory, 14 Kadison conjecture, 37 Künneth spectral sequence, 77

lattice, 10 complete, 10, 11, 16 of projections, 11

lattice isomorphism, 10, 11 lattice of Hilbert A-modules, 11 Laurent polynomial ring, 3

skew-, 63, 64 left exact functor, 20 left full

definition of, 108 left regular representation, 10 Linnell

’s class of groups, 45 Linnell’s class of group, 45 Linnell’s class of groups and DΓ, 52 and RΓ, 52 and Tor_{p}^{ℂΓ}(−; DΓ), 77 and Euler characteristics, 78 and free products, 46 examples, 46

Linnell’s theorem, 52 local homomorphism, 105 localization

and K-theory, 85 and crossed products, 56 and extensions, 58 iterated, 59 Ore, 8, 64 sequence, 85

lower multiplicatively closed, 101

map

stabilization, 14 matrices

stably associated, 60 matrix algebra, 14 matrix ring

and Ore localization, 100 of a von Neumann regular ring,

95 mean

invariant, 44 measurable function, 6 module, 4

arbitrary, 21 finitely generated projective, 20 finitely presented, 20 Fredholm, 37, 38 Hilbert A-, 10 torsion, 19, 21, 22, 85 torsionfree, 19

modules finitely generated free, 101 finitely generated projective, 101 Moody’s induction theorem, 36, 70 Murray von Neumann equivalent,

13

negative K-theory, 34 noetherian ring, 64, 114

and crossed products, 62 non-zerodivisor, 4, 7 in UG ∗ ℤ, 64 non-zerodivisors of ℂΓ, 52 normal trace, 5, 15 normalization of the trace, 15 Novikov-Shubin invariants, 25

operator affiliated, 89 closed, 87 core of, 88 domain of, 87 finite rank, 37 graph of, 87 invertible, 7, 84 selfadjoint, 88 spectrum of, 88 symmetric, 88 trace class, 37 unbounded, 87 unique closure of, 94

operators

affiliated to a group von Neumann algebra, 90

orbit category, 33

tensor product over, 34 Ore

condition, 8

localization, 8 Ore condition

definition of, 98 for ℂΓ, 53

Ore localization and G-theory, 69 and matrix rings, 100 and noetherian semiprime rings,

64 definition of, 99 is exact, 99 of ℂΓ, 53

polar decomposition, 89 polycyclic-by-finite group

and crossed product, 62 polynomial

Laurent, 3 prime ring

definition of, 114 processes

C closed under, 46 product

free, 49 of unbounded operators, 6, 87

projection, 4, 12, 13

valued measure, 88 projections

isomorphism classes of, 15 isomorphism of, 13 lattice of, 11 Murray von Neumann equiva-

lence of, 13 projective dimension, 85 projective rank function, 76

definition of, 108 push-out

of rings, 66, 100 rank

Hattori-Stallings, 35 rank function

faithful, 108 monotone, 108 over semihereditary ring, 109 projective, 76, 108 Sylvester projective, 76

rational closure, 32, 52

definition of, 103 rationally closed

definition of, 103 reduced L^{2}-homology, 24 representation

left regular, 10 very trivial, 40

resolvent set, 88 right full

definition, 108 ring, 4

∗-regular, 9 artinian, 62, 114 coherent, 11, 20, 114 division, 105 hereditary, 76, 109 homomorphism, 4 noetherian, 64, 114 prime, 114 semihereditary, 11, 20, 30, 114 semiprime, 62, 64, 114 semisimple, 31, 62, 114 unit regular, 8, 13 von Neumann regular, 6, 11,

29, 95, 114 ring homomorphism

X-inverting, 98 Σ-inverting, 101 universal Σ-inverting, 101

universal X-inverting, 98 ring of fractions, 99

classical, 99 definition of, 105

rule Cramer’s, 60, 85, 102

saturation, 101 Schatten ideals, 37 selfadjoint operator, 88 semifir, 75, 106

and universal field of fractions,

75 definition of, 114 examples of, 106 universal field of fractions of,

106 semihereditary ring, 11, 20, 30

and finitely presented modules,

86 definition of, 114

semiprime

subring of UΓ, 62 semiprime ring, 62, 64

definition of, 114 semiprime rings

and crossed products, 62 semisimple ring, 31, 62, 114

structure theory of, 64 Serre’s conjecture, 28 skew field, 105 skew-Laurent polynomial ring, 63,

64 specialization, 105 spectral sequence

Atiyah-Hirzebruch, 33 Künneth, 77

spectrum gap in, 84 of an operator, 88

square integrable function, 6 stabilization map, 14 stably associated, 60, 102 step function, 82 Stieltjes integral, 89 strong topology, 5 subrings

∗-closed, 62 subspace

affiliated, 89

essentially dense, 91 sum

of unbounded operators, 6, 87 summability

of Fredholm module, 38 symmetric operator, 88

tensor product over orbit category,

34 theorem

Goldie’s, 64 topology

strong, 5 ultraweak, 5 weak, 5

Tor-dimension, 6 torsion

module, 19 torsion element, 20 torsion module, 21, 22 torsionfree group, 26, 31

and Atiyah conjecture, 32 torsionfree module, 19 trace, 5, 91

center valued, 81, 82

faithful, 5, 15 finite, 5 normal, 5, 15 normalization of, 15 of a group von Neumann alge-

bra, 5 transfinite induction, 43, 46 tree, 39

ultraweak topology, 5 unbounded operator

commuting with bounded, 89 definition of, 87 domain of, 87 extension of, 87 graph of, 87 spectrum of, 88

unbounded operators composition of, 87 product of, 87 sum of, 87

union

directed, 4, 71 unit regular ring, 8, 13

K_{0} of, 97 K_{1} of, 84 definition of, 96

units

abelianized group of, 84 universal

Σ(Γ)-inverting, 52 T(Γ)-inverting, 53

universal Σ-inverting ring homomor-

phism, 101 universal R-field

definition of, 105 universal coefficient theorem

for L^{2}-homology, 77 universal field of fractions definition of, 105 of a semifir, 75, 106 recognizing, 107

universal localization K-theory of, 76 and K-theory, 85 of a commutative ring, 102

unreduced L^{2}-homology, 24

very trivial representation, 40 virtual Euler characteristic, 28 virtually abelian group, 81 virtually cyclic group, 36 von Neumann algebra, 5

K_{0} of, 81 K_{1} of, 84 amenable, 45 center of, 81 finite, 5 of a group, 1 of the group ℤ, 3 of type I_{n}, 81 of virtually abelian group, 81 type of, 81

von Neumann regular ring, 6, 11,

29, 114 and chain conditions, 96 and directed union, 71, 95 and matrix ring, 95 and zerodivisors, 96 center of, 95 definition of, 95 is division closed, 103 is rationally closed, 103

weak dimension, 6 weak isomorphism, 7, 83 weak topology, 5 zero divisor conjecture, 28 zerodivisor, 4, 98

conjecture, 28

in von Neumann regular ring, 96

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