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 $C^{\ast }$-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 $\mathrm{Tor}({\bar{E}}_D^1(X))$.

14 pages, this version agrees with the one that will be published in the Journal of Noncommutative Geometry

References in corpus (3)

• Strongly self-absorbing C*-algebras are Z-stable
• A Dixmier-Douady theory for strongly self-absorbing C*-algebras
• C_0(X)-algebras, stability and strongly self-absorbing C*-algebras

Cited by in corpus (8)

• Poly-$\mathbb{Z}$ group actions on Kirchberg algebras II
• Classification of ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras
• Equivariant higher Dixmier-Douady Theory for circle actions on UHF-algebras
• Bundles of strongly self-absorbing $C^{\ast }$-algebras with a Clifford grading
• A generalization of the topological Brauer group
• A weak homotopy equivalence type result related to Kirchberg algebras
• A topological invariant for continuous fields of Cuntz algebras II
• Computations in higher twisted $K$-theory