Charlotte Wahl
A noncommutative index theorem \\ for a manifold with boundary \\ and cylindric ends \\ Part I
Preprint series: Mathematica Gottingensis

MSC:

46L89 Other ``noncommutative'' mathematics based on $C^*$-algebra theory, See also {58B30, 58G12}

19K56 Index theory, See also {58G12}

19D55 $K$-theory and homology; cyclic homology and cohomology, See also {18G60}

Abstract: We generalize the Maslov index introduced by Kashiwara to a triple of
pairwise transverse Lagrangian submodules of $A^{2n}$ where $A$ is a $C^*$-algebra
and $A^{2n}$ is endowed with a nondegenerated skewhermitian form.
Our Maslov index is an element of the $K$-theory of $A$.
Furthermore we define a noncommutative eta-form associated to a pair
of Lagrangian submodules of $A^{2n}$. Then we prove a formula expressing
the Chern character of the Maslov index in terms of eta-forms in the de Rham
homology of certain dense subalgebras of $A$. The formula and its proof are modelled
on a result of Bunke and Koch who studied the family case: The Maslov index
is the index of a Dirac operator on a two dimensional manifold with boundary
and cylindric ends. The noncommutative index problem is solved by adapting
the superconnection formalism. For the construction and the study of the heat kernels
the theory of holomorphic semigroups is used.

This is part I, part II will appear in Mathematica Gottingensis 12

Keywords: noncommutative index theory, eta-forms, Maslov index, boundary value problems
Notes: Sonderdruck