Duality theory for nonergodic actions

Generalizing work by Pinzari and Roberts, we characterize actions of a compact quantum group G on C*-algebras in terms of what we call weak unitary tensor functors from Rep G into categories of C*-correspondences. We discuss the relation of our construction of a C*-algebra from a functor to some well-known crossed product type constructions, such as cross-sectional algebras of Fell bundles and crossed products by Hilbert bimodules. We also relate our setting to recent work of De Commer and Yamashita by showing that any object in a module C*-category over Rep G produces a weak unitary tensor functor, and, as a consequence, actions can also be described in terms of (Rep G)-module C*-categories. As an application we discuss deformations of C*-algebras by cocycles on discrete quantum groups.