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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Pull-backs of $ C\sp \ast$-algebras and crossed products by certain diagonal actions


Authors: Iain Raeburn and Dana P. Williams
Journal: Trans. Amer. Math. Soc. 287 (1985), 755-777
MSC: Primary 46L05; Secondary 46L40, 46L55
DOI: https://doi.org/10.1090/S0002-9947-1985-0768739-2
MathSciNet review: 768739
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Abstract: Let $ G$ be a locally compact group and $ p:\Omega \to T$ a principal $ G$-bundle. If $ A$ is a $ {C^\ast}$-algebra with primitive ideal space $ T$, the pull-back $ {p^\ast}A$ of $ A$ along $ p$ is the balanced tensor product $ {C_0}(\Omega ){ \otimes _{C(T)}}A$. If $ \beta :G \to \operatorname{Aut}\,A$ consists of $ C(T)$-module automorphisms, and $ \gamma :G \to \operatorname{Aut}\,{C_0}(\Omega )$ is the natural action, then the automorphism group $ \gamma \otimes \beta $ of $ {C_0}(\Omega ) \otimes A$ respects the balancing and induces the diagonal action $ {p^\ast}\beta $ of $ G$ on $ {p^\ast}A$. We discuss some examples of such actions and study the crossed product $ {p^\ast}A{ \times _{{p^\ast}\beta }}G$. We suggest a substitute $ D$ for the fixed-point algebra, prove $ {p^\ast}A \times G$ is strongly Morita equivalent to $ D$, and investigate the structure of $ D$ in various cases. In particular, we ask when $ D$ is strongly Morita equivalent to $ A$--sometimes, but by no means always--and investigate the case where $ A$ has continuous trace.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0768739-2
Keywords: Crossed products, principal bundles, imprimitivity bimodules, strong Morita equivalence, locally unitary automorphism group, Dixmier-Douady class, twisted transformation group $ {C^\ast}$-algebra
Article copyright: © Copyright 1985 American Mathematical Society