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

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 ${\text {Au}}{{\text {t}}_{C(T)}}A$.