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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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