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

TL;DR

This paper extends the Dixmier-Douady classification to continuous fields of C*-algebras with fibers involving strongly self-absorbing algebras, introducing a generalized cohomology framework with characteristic classes.

Contribution

It develops a generalized cohomology theory for classifying continuous fields with fibers A⊗K, where A is strongly self-absorbing, expanding the classical Dixmier-Douady theory.

Findings

Classification involves a computable generalized cohomology theory.

Characteristic classes appear in higher dimensions.

Provides K-theoretical criteria for local triviality.

Abstract

We show that the Dixmier-Douady theory of continuous field $C^*$-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.