The structure of the Brauer group and crossed products of $C_0(X)$-linear group actions on $C_0(X,\mathcal K)$

Abstract:

For a second countable locally compact group $G$ and a second countable locally compact space $X$ let $\operatorname {Br}_G(X)$ denote the equivariant Brauer group (for the trivial $G$-space $X$) consisting of all Morita equivalence classes of spectrum fixing actions of $G$ on continuous-trace $C^*$-algebras $A$ with spectrum $\widehat {A}=X$. Extending recent results of several authors, we give a complete description of $\operatorname {Br}_G(X)$ in terms of group cohomology of $G$ and Čech cohomology of $X$. Moreover, if $G$ has a splitting group $H$ in the sense of Calvin Moore, we give a complete description of the $C_0(X)$-bundle structure of the crossed product $A\rtimes _{\alpha }G$ in terms of the topological data associated to the given action $\alpha :G\to \operatorname {Aut} A$ and the bundle structure of the group $C^*$-algebra $C^*(H)$ of $H$.