The structure of the Brauer group and crossed products of $C_0(X)$-linear group actions on $C_0(X,\mathcal K)$
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- by Siegfried Echterhoff and Ryszard Nest PDF
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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 Čech 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 $H$.References
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Additional Information
- Siegfried Echterhoff
- Affiliation: Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany
- MR Author ID: 266728
- ORCID: 0000-0001-9443-6451
- Email: echters@math.uni-muenster.de
- Ryszard Nest
- Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark
- MR Author ID: 130350
- Email: rnest@math.ku.dk
- Received by editor(s): March 9, 1999
- Received by editor(s) in revised form: March 3, 2000
- Published electronically: May 4, 2001
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1837255