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

Abstract

Abstract We show that the Dixmier–Douady theory of continuous fields of C * C^{*} -algebras with compact operators 𝕂 {\mathbb{K}} as fibers extends significantly to a more general theory of fields with fibers A ⊗ \otimes 𝕂 \mathbb{K} where A is a strongly self-absorbing C * 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.