# A Dixmier–Douady theory for strongly self-absorbing $C_{∗}$-algebras II: the Brauer group

### Marius Dadarlat

Purdue University, West Lafayette, United States### Ulrich Pennig

Universität Münster, Germany

## 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 $D⊗K$, 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 $Eˉ_{D}(X)$ 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 $Eˉ_{D}(X)$ arise from locally trivial fields with fiber $D⊗M_{n}(C)$, $n≥1$, for all known examples of strongly self-absorbing $C_{∗}$ -algebras} $D$. Moreover the Brauer group generated by locally trivial fields with fiber $D⊗M_{n}(C)$, $n≥1$ is isomorphic to $TorEˉ_{D}(X)$.

## Cite this article

Marius Dadarlat, Ulrich Pennig, A Dixmier–Douady theory for strongly self-absorbing $C_{∗}$-algebras II: the Brauer group. J. Noncommut. Geom. 9 (2015), no. 4, pp. 1137–1154

DOI 10.4171/JNCG/218