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The structure of the Brauer group and crossed products of $C_0(X)$-linear group actions on $C_0(X,\mathcal K)$


Authors: Siegfried Echterhoff and Ryszard Nest
Journal: Trans. Amer. Math. Soc. 353 (2001), 3685-3712
MSC (1991): Primary 46L55; Secondary 22D25
DOI: https://doi.org/10.1090/S0002-9947-01-02794-5
Published electronically: May 4, 2001
MathSciNet review: 1837255
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Abstract:

For a second countable locally compact group $G$ and a second countable locally compact space $X$let $\operatorname{Br}_G(X)$ denote the equivariant Brauer group (for the trivial $G$-space $X$) consisting of all Morita equivalence classes of spectrum fixing actions of $G$ on continuous-trace $C^*$-algebras $A$ with spectrum $\widehat{A}=X$. Extending recent results of several authors, we give a complete description of $\operatorname{Br}_G(X)$ in terms of group cohomology of $G$ and Cech cohomology of $X$. Moreover, if $G$ has a splitting group $H$ in the sense of Calvin Moore, we give a complete description of the $C_0(X)$-bundle structure of the crossed product $A\rtimes_{\alpha}G$ in terms of the topological data associated to the given action $\alpha:G\to \operatorname{Aut} A$and the bundle structure of the group $C^*$-algebra $C^*(H)$ of $H$.


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Additional Information

Siegfried Echterhoff
Affiliation: Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany
Email: echters@math.uni-muenster.de

Ryszard Nest
Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
Email: rnest@math.ku.dk

DOI: https://doi.org/10.1090/S0002-9947-01-02794-5
Received by editor(s): March 9, 1999
Received by editor(s) in revised form: March 3, 2000
Published electronically: May 4, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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