Bundles of C*-categories, II: C*-dynamical systems and Dixmier-Douady invariants

TL;DR

This paper introduces a cohomological invariant that generalizes the Dixmier-Douady class, capturing obstructions in C*-algebra bundles related to gauge actions and duality in tensor categories.

Contribution

It defines a new nonabelian cohomological invariant for C*-algebra bundles, extending the Dixmier-Douady class and linking it to duality and embedding problems in C*-algebra theory.

Findings

The invariant generalizes the Dixmier-Douady class.

It characterizes obstructions to embedding into Cuntz-Pimsner algebras.

Application to gauge-equivariant K-theory and duality breaking.

Abstract

We introduce a cohomological invariant arising from a class in nonabelian cohomology. This invariant generalizes the Dixmier-Douady class and encodes the obstruction to a C*-algebra bundle being the fixed-point algebra of a gauge action. As an application, the duality breaking for group bundles vs. tensor C*-categories with non-simple unit is discussed in the setting of Nistor-Troitsky gauge-equivariant K-theory: there is a map assigning a nonabelian gerbe to a tensor category, and "triviality" of the gerbe is equivalent to the existence of a dual group bundle. At the C*-algebraic level, this corresponds to studying C*-algebra bundles with fibre a fixed-point algebra of the Cuntz algebra and in this case our invariant describes the obstruction to finding an embedding into the Cuntz-Pimsner algebra of a vector bundle.