Bundles of \(C^*\)-categories. II: \(C^*\)-dynamical systems and Dixmier-Douady invariants (Q1029312)

The contents of this paper are as follows: 1 Introduction, 2 Preliminaries, 3 Tensor \(C^*\)-categories and \(C^*\)-dynamical systems, 4 Cohomology classes and principal bundles, 5 Bundles of \(C^*\)-algebras and cohomology classes, 6 Gauge-equivariant bundles and a concrete duality, 7 Cohomological invariants and duality breaking. As mentioned in Section~1, a tensor \(C^*\)-category is viewed as the dual object of a compact group by [\textit{S.\,Doplicher} and \textit{J.\,E.\thinspace Roberts}, J.~Funct.\ Anal.\ 74, 96--120 (1987; Zbl 0619.46053); Invent.\ Math.\ 98, No.\,1, 157--218 (1989; Zbl 0691.22002)], who discovered a duality theory for compact groups in the \(C^*\)-dynamical systems of Cuntz algebras \(O_n\) by compact subgroups \(G\) of \(U(n)\). The assignment from \(G\) to this \(C^*\)-dynamical system is viewed as Galois correspondence. It has been proved by the author in Part~I of this paper [J.~Funct.\ Anal.\ 247, No.\,2, 351--377 (2007; Zbl 1126.46048)] that every \(C^*\)-category with symmetry and conjugates and with nonsimple unit, i.e., the space of arrows of the identity object isomorphic to \(C(X)\), can be regarded as a bundle of \(C^*\)-categories over \(X\) with fibers duals of compact groups, and also, locally trivial \(C^*\)-dynamical systems over \(X\) with fibers such Cuntz \(C^*\)-dynamical systems are classified in terms of the \(1\)-cohomology set over \(X\). The Galois correspondence when \(X\) is nontrivial is studied in this paper. The Cuntz-Pimsner algebra associated with the module of sections of a vector bundle over \(X\) yields a new \(C^*\)-dynamical system. This leads to another duality in the sense of \textit{V.\,Nistor} and \textit{E.\,Troitsky} [Trans.\ Am.\ Math.\ Soc.\ 356, No.\,~1, 185--218 (2004; Zbl 1030.46095)]. A cohomological obstruction yields new phenomena that do not arise in the Doplicher-Roberts construction when \(X\) is trivial. The obstruction giving isomorphic \(C^*\)-dynamcal systems with non-isomorphic bundles has roots in the general framework of principal bundles, and that obstruction and the Dixmier-Douady invariant for bundles with fibers the \(C^*\)-algebra of compact operators are particular cases are applied in \(C^*\)-algebra bundles.