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

We introduce a cohomological invariant arising from a class in nonabelian cohomology. This invariant generalises 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 nonsimple 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.