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

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}$)$,$n\geq 1$, 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 $\mathrm{T}\mathrm{o}\mathrm{r}({\bar{E}}_D^1(X))$.