H. Baumgaertel | Searcharxiv

arXiv2000

Hilbert C*-systems for actions of the circle group

The paper contains constructions of Hilbert systems for the action of the circle group $T$ using subgroups of implementable Bogoljubov unitaries w.r.t. Fock representations of the Fermion algebra for suitable data of the selfdual framework: ${\cal H}$ is the reference Hilbert space, $Γ$ the conjugation and $P$ a basis projection on ${\cal H}.$ The group $C({spec} {\cal Z}\to T)$ of $T$-valued functions on ${spec} {\cal Z}$ turns out to be isomorphic to the stabilizer of ${\cal A}$. In particular, examples are presented where the center ${\cal Z}$ of the fixed point algebra ${\cal A}$ can be calculated explicitly.

math-ph↗

arXiv2000

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

Given a C*-algebra $A$, a discrete abelian group $X$ and a homomorphism $Θ: X\to$ Out$A$ defining the dual action group $Γ\subset$ aut$A$, the paper contains results on existence and characterization of Hilbert $\{A,Γ\}$, where the action is given by $\hat{X}$. 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 [Mem.Am.Math.Soc. 28 Nr 237 (1980)] or Sutherland [Publ.Res.Inst.Math.Sci. 16 (1980) 135]. A Hilbert extension exists iff there is a generalized 2-cocycle. These results generalize those in [Commun.Math.Phys. 15 (1969) 173], 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 $H^2(X,{\cal U}(Z),α_X)$ 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 Sutherland 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.

math.OA↗