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

Abstract

We show that the Dixmier-Douady theory of continuous fields of C∗-algebras with compact operators K as fibers extends significantly to a more general theory of fields with fibers A ⊗ 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.