A Dixmier–Douady theory for strongly self-absorbing $C^{\ast }$-algebras II: the Brauer group

Abstract

We have previously shown that the isomorphism classes of orientable locally trivial fields of $C^{\ast }$ -algebras over a compact metrizable space $X$ with fiber $D\otimes \mathbb{K}$, where $D$ is a strongly self-absorbing $C^{\ast }$ -algebra, form an abelian group under the operation of tensor product. Moreover this group is isomorphic to the first group ${\bar{E}}_D^1(X)$ 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 ${\bar{E}}_D^1(X)$ arise from locally trivial fields with fiber $D\otimes M_n(\mathbb{C})$, $n\ge 1$, for all known examples of strongly self-absorbing $C^{\ast }$ -algebras} $D$. Moreover the Brauer group generated by locally trivial fields with fiber $D\otimes M_n(\mathbb{C})$, $n\ge 1$ is isomorphic to $Tor{\bar{E}}_D^1(X)$.