Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 445

Dual group actions on C*-algebras and their description by Hilbert extensions.

H. Baumgaertel, F. Lledó

Given a C*-algebra ${cal A}$, a discrete abelian group ${cal X}$ and a homomorphism $Thetacolon {cal X} ightarrow mbox{Out},{cal A},$ defining the dual action group $Gammasubset mbox{aut},{cal A}$, the paper contains results on existence and characterization of Hilbert extensions of ${cal A},Gamma}$, where the action is given by $hat{cal X}$. They are stated at the (abstract) C*-level and can therefore be considered as a refinement of the extension results given for von Neumann algebras for example by Jones [16] or Sutherland [20,21]. A Hilbert extension exists iff there is a generalized 2-cocycle. These results generalize those in [10], which are formulated in the context of superselection theory, where it is assumed that the algebra $al A.$ has a trivial center, i.e.~${cal Z}=CEINS$. In particular the well-known ``outer characterization'' of the second cohomology $H^{2}({cal X},{cal U}({cal Z}),alpha_{cal X})$ can be reformulated: there is a bijection to the set of all ${cal A}$-module isomorphy classes of Hilbert extensions.Finally, a Hilbert space representation (due to Sutherland [20,21] in the von Neumann case) is mentioned. The C*-norm of the Hilbert extension is expressed in terms of the norm of this representation and it is linked to the so-called regular representation appearing in superselection theory.

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