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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Pull-backs of $C^ \ast$-algebras and crossed products by certain diagonal actions
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by Iain Raeburn and Dana P. Williams PDF
Trans. Amer. Math. Soc. 287 (1985), 755-777 Request permission

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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Additional Information
  • © Copyright 1985 American Mathematical Society
  • 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