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 [Formula: see text] be the rigid C*-tensor category of finite-dimensional continuous unitary representations of G. Let [Formula: see text] be the rigid C*-2-category of cofinite semisimple finitely decomposable [Formula: see text]-module categories. We show that finite-dimensional G- C*-algebras can be identified with equivalence classes of 1-morphisms out of the object [Formula: see text] in [Formula: see text]. For 1-morphisms [Formula: see text], [Formula: see text], we show that covariant completely positive maps between the corresponding G- C*-algebras can be “dilated” to isometries τ: X → Y ⊗ E, where [Formula: see text] is some “environment” 1-morphism. Dilations are unique up to partial isometry on the environment; in particular, the dilation minimizing 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.