A covariant Stinespring theorem

Abstract

We prove a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be a compact quantum group, and let T:= Rep(G) be the rigid C*-tensor category of finite-dimensional continuous unitary representations of G. Let Mod(T) be the rigid C*-2-category of cofinite semisimple finitely decomposable T-module categories. We show that finite-dimensional G-C*-algebras can be identified with equivalence classes of 1-morphisms out of the object T in Mod(T). For 1-morphisms X: T -> M1, Y: T -> M2, we show that covariant completely positive maps between the corresponding G-C*-algebras can be 'dilated' to isometries t: X -> Y \otimes E, where E: M2 -> M1 is some 'environment' 1-morphism. Dilations are unique up to partial isometry on the environment; in particular, the dilation minimising the quantum dimension of the environment is unique up to a unitary. When G is a compact group this recovers previous covariant Stinespring-type theorems.