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

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

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0013-54FF-5 Version Permalink: http://hdl.handle.net/11858/00-001M-0000-0013-5500-8
Genre: Journal Article

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 Creators:
Baumgärtel, Hellmut, Author
Lledo, Fernando1, Author
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1Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, escidoc:24012              

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 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.

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Language(s): eng - English
 Dates: 2002
 Publication Status: Published in print
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 Identifiers: eDoc: 2791
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Title: Mathematische Nachrichten
Source Genre: Journal
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Pages: - Volume / Issue: 239-240 Sequence Number: - Start / End Page: 11 - 27 Identifier: -