Pull-backs of $C^ \ast$-algebras and crossed products by certain diagonal actions

Abstract:

Let $G$ be a locally compact group and $p:\Omega \to T$ a principal $G$-bundle. If $A$ is a ${C^\ast }$-algebra with primitive ideal space $T$, the pull-back ${p^\ast }A$ of $A$ along $p$ is the balanced tensor product ${C_0}(\Omega ){ \otimes _{C(T)}}A$. If $\beta :G \to \operatorname {Aut} A$ consists of $C(T)$-module automorphisms, and $\gamma :G \to \operatorname {Aut} {C_0}(\Omega )$ is the natural action, then the automorphism group $\gamma \otimes \beta$ of ${C_0}(\Omega ) \otimes A$ respects the balancing and induces the diagonal action ${p^\ast }\beta$ of $G$ on ${p^\ast }A$. We discuss some examples of such actions and study the crossed product ${p^\ast }A{ \times _{{p^\ast }\beta }}G$. We suggest a substitute $D$ for the fixed-point algebra, prove ${p^\ast }A \times G$ is strongly Morita equivalent to $D$, and investigate the structure of $D$ in various cases. In particular, we ask when $D$ is strongly Morita equivalent to $A$—sometimes, but by no means always—and investigate the case where $A$ has continuous trace.