English
 
Help Privacy Policy Disclaimer
  Advanced SearchBrowse

Item

ITEM ACTIONSEXPORT

Released

Journal Article

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

MPS-Authors

Lledo,  Fernando
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

External Resource
No external resources are shared
Fulltext (restricted access)
There are currently no full texts shared for your IP range.
Fulltext (public)

2791.pdf
(Preprint), 203KB

Supplementary Material (public)
There is no public supplementary material available
Citation

Baumgärtel, H., & Lledo, F. (2002). Dual Group Actions on C*-Algebras and Their Description by Hilbert Extensions. Mathematische Nachrichten, 239-240, 11-27.


Cite as: https://hdl.handle.net/11858/00-001M-0000-0013-54FF-5
Abstract
Given a C*-algebra $A$, a discrete abelian group $X$ and a homomorphism $Theta: X o$ Out$A$ defining the dual action group $Gammasubset$ aut$A$, the paper contains results on existence and characterization of Hilbert ${A,Gamma}$, 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 $H^2(X,{cal U}(Z),alpha_X)$ 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.