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On the structure of twisted group $ C\sp *$-algebras


Authors: Judith A. Packer and Iain Raeburn
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
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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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DOI: https://doi.org/10.1090/S0002-9947-1992-1078249-7
Article copyright: © Copyright 1992 American Mathematical Society

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