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 $ \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 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.

Massless Scalar Free Field in 1+1 Dimensions I: Weyl Algebras Products and Superselection Sectors

“…is a short exact sequence. 4 This means that the discrete twisted crossed product U(V, σ V ) := T ⋊ (ι,y) U(V ) is an extension of the Abelian formal symbols group U(V ) on the symplectic space V , by the torus group T and the 2-cocycle (see [47])…”

“…where the action is trivial, β ≡ ι, and the function y(v, v ′ ) := e − i 2 σV (v,v ′ ) is defined by the symplectic form. Hence, the above announced fourth category is defined by 4 Given a group G an extension E of it by another group N is described by the short exact sequence…”

“…Actually this isomomorphism corresponds to a section G ∋ (c, q) → ∆ c /Inn A, where ∆ c is the group of the covariant DHR automorphisms and Inn A the one of the inner automorphisms of A. This section gives a group homomorphism, as said in [19, Section IV] and a similar choice is also made in a non trivial center situation, as illustrated in [4].…”

Superselection in the presence of constraints

“…[2] for details). For the case Z(A) ⊃ C 1 l see [22] (though the minimal case is not mentioned there).…”