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

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- by Iain Raeburn
- Trans. Amer. Math. Soc.
**263**(1981), 183-205 - DOI: https://doi.org/10.1090/S0002-9947-1981-0590419-3
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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 ${\text {Au}}{{\text {t}}_{C(T)}}A$.## References

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## Bibliographic Information

- © Copyright 1981 American Mathematical Society
- 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