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A Dixmier-Douady theory for strongly self-absorbing $$C^\ast$$-algebras. (English) Zbl 1431.46051
Summary: We show that the Dixmier-Douady theory of continuous fields of $$C^\ast$$-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 $$C^\ast$$-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.

##### MSC:
 46L80 $$K$$-theory and operator algebras (including cyclic theory) 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 55N15 Topological $$K$$-theory
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