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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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On the structure of twisted group $C^ *$-algebras
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by Judith A. Packer and Iain Raeburn PDF
Trans. Amer. Math. Soc. 334 (1992), 685-718 Request permission

Abstract:

We first give general structural results for the twisted group algebras ${C^{\ast } }(G,\sigma )$ of a locally compact group $G$ with large abelian subgroups. In particular, we use a theorem of Williams to realise ${C^{\ast }}(G,\sigma )$ as the sections of a ${C^{\ast }}$-bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras. Next we use a device of Rosenberg to show that, when $\Gamma$ is a discrete subgroup of a solvable Lie group $G$, the $K$-groups ${K_ {\ast } }({C^{\ast } }(\Gamma ,\sigma ))$ are isomorphic to certain twisted $K$-groups ${K^{\ast } }(G/\Gamma ,\delta (\sigma ))$ of the homogeneous space $G/\Gamma$, and we discuss how the twisting class $\delta (\sigma ) \in {H^3}(G/\Gamma ,\mathbb {Z})$ depends on the cocycle $\sigma$. For many particular groups, such as ${\mathbb {Z}^n}$ or the integer Heisenberg group, $\delta (\sigma )$ always vanishes, so that ${K_ {\ast } }({C^{\ast } }(\Gamma ,\sigma ))$ is independent of $\sigma$, but a detailed analysis of examples of the form ${\mathbb {Z}^n} \rtimes \mathbb {Z}$ shows this is not in general the case.
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Additional Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 334 (1992), 685-718
  • MSC: Primary 22D25; Secondary 19K99, 46L55, 46L80
  • DOI: https://doi.org/10.1090/S0002-9947-1992-1078249-7
  • MathSciNet review: 1078249