Dual Group Actions on C*-Algebras and Their Description by Hilbert Extensions

Abstract

Given a C*-algebra 𝒜, a discrete abelian group 𝒳 and a homomorphism Θ : 𝒳 → Out 𝒜, defining the dual action group Γ ⊂ aut 𝒜, the paper contains results on existence and characterization of Hilbert extensions of {𝒜, Γ}, where the action is given by . 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 V. F. R. Jones [18] or C. E. Sutherland [22, 23]. A Hilbert extension exists iff there is a generalized 2-cocycle. These results generalize those in [12], which are formulated in the context of superselection theory, where it is assumed that the algebra 𝒜 has a trivial center, i.e. 𝒵 = ℂ1. In particular the well-known “outer characterization” of the second cohomology H2(𝒳,𝒰(𝒵),α𝒳) can be reformulated: there is a bijection to the set of all 𝒜-module isomorphy classes of Hilbert extensions. Finally, a Hilbert space representation (due to C. E. Sutherland [22, 23] 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.