Journal of Noncommutative Geometry


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Volume 9, Issue 4, 2015, pp. 1137–1154
DOI: 10.4171/JNCG/218

Published online: 2016-01-06

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

Marius Dadarlat[1] and Ulrich Pennig[2]

(1) Purdue University, West Lafayette, United States
(2) Universität Münster, Germany

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 $\bar{E}^1_D (X)$ 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^\ast $-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 $T or {\bar{E}^1_D(X)}$.

Keywords: Strongly self-absorbing, $^\ast$-algebras, Dixmier–Douady class, Brauer group, torsion, opposite algebra

Dadarlat Marius, Pennig Ulrich: A Dixmier–Douady theory for strongly self-absorbing $C^\ast$-algebras II: the Brauer group. J. Noncommut. Geom. 9 (2015), 1137-1154. doi: 10.4171/JNCG/218