A Dixmier-Douady Theory for strongly self-absorbing C*-algebras II: the Brauer group

Abstract

We have previously shown that the isomorphism classes of orientable locally trivial fields of $C^*$-algebras over a compact metrizable space $X$ with fiber $D\otimes \mathbb{K}$, where $D$ is a strongly self-absorbing $C^*$-algebra, form an abelian group under the operation of tensor product. Moreover this group is isomorphic to the first group $\bar{E}^1_D(X)$ of the (reduced) generalized cohomology theory associated to the unit spectrum of topological K-theory with coefficients in $D$. Here we show that all the torsion elements of the group $\bar{E}^1_D(X)$ arise from locally trivial fields with fiber $D \otimes M_n(\mathbb{C})$, $n\geq 1$, for all known examples of strongly self-absorbing $C^*$-algebras $D$. Moreover the Brauer group generated by locally trivial fields with fiber $D\otimes M_n(\mathbb{C})$, $n\geq 1$ is isomorphic to ${\rm Tor}(\bar{E}^1_D(X))$.