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Transactions of the American Mathematical Society

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

 

 

On the Picard group of a continuous trace $ C\sp{\ast} $-algebra


Author: Iain Raeburn
Journal: Trans. Amer. Math. Soc. 263 (1981), 183-205
MSC: Primary 46M20; Secondary 18F25, 46L05, 58G12
DOI: https://doi.org/10.1090/S0002-9947-1981-0590419-3
MathSciNet review: 590419
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Abstract: Let $ A$ be a continuous trace $ {C^\ast}$-algebra with paracompact spectrum $ T$, and let $ C(T)$ be the algebra of bounded continuous functions on $ T$, so that $ C(T)$ acts on $ A$ in a natural way. An $ A - A$ bimodule $ X$ is an $ A{ - _{C(T)}}A$ imprimitivity bimodule if it is an $ A - A$ imprimitivity bimodule in the sense of Rieffel and the induced actions of $ C(T)$ on the left and right of $ X$ agree. We denote by $ {\text{Pi}}{{\text{c}}_{C(T)}}A$ the group of isomorphism classes of $ A{ - _{C(T)}}A$ imprimitivity bimodules under $ { \otimes _A}$. Our main theorem asserts that $ {\text{Pi}}{{\text{c}}_{C(T)}}A \cong {\text{Pi}}{{\text{c}}_{C(T)}}{C_0}(T)$. This result is well known to algebraists if $ A$ is an $ n$-homogeneous $ {C^\ast}$-algebra with identity, and if $ A$ is separable it can be deduced from two recent descriptions of the automorphism group $ {\text{Au}}{{\text{t}}_{C(T)}}A$ due to Brown, Green and Rieffel on the one hand and Phillips and Raeburn on the other. Our main motivation was to provide a direct link between these two characterisations of $ {\text{Au}}{{\text{t}}_{C(T)}}A$.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0590419-3
Article copyright: © Copyright 1981 American Mathematical Society