A Dixmier-Douady theory for strongly self-absorbing C*-algebras

Abstract

We show that the Dixmier-Douady theory of continuous field $C^{\ast }$-algebras with compact operators $\mathbb{K}$ as fibers extends significantly to a more general theory of fields with fibers $A\otimes \mathbb{K}$ where $A$ is a strongly self-absorbing C*-algebra. The classification of the corresponding locally trivial fields involves a generalized cohomology theory which is computable via the Atiyah-Hirzebruch spectral sequence. An important feature of the general theory is the appearance of characteristic classes in higher dimensions. We also give a necessary and sufficient $K$-theoretical condition for local triviality of these continuous fields over spaces of finite covering dimension.

arXiv comment26p, this version agrees with the one published in J. Reine Angew. Math. except for a minor correction and an extension of Corollary 4.8, both of which were done after publication

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• Strongly self-absorbing C*-algebras are Z-stable
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• Colocalizations of noncommutative spectra and bootstrap categories
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• Loop group actions on Cuntz algebras and geometric higher twists
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• A weak homotopy equivalence type result related to Kirchberg algebras
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