Functoriality of the KSGNS Construction for Intertwiners of Strict Positive $C^{\ast }$-Correspondences

Abstract

We prove that the KSGNS construction can be viewed as an endofunctor on a category whose objects are strict positive $C^{\ast }$-correspondences from a fixed $C^{\ast }$-algebra and morphisms are given by intertwiners which account for automorphisms of the fixed $C^{\ast }$-algebra. Using this perspective, we provide a functorial perspective for strict positive equivariant $C^{\ast }$-correspondences of $C^{\ast }$-dynamical systems and show every strict positive equivariant $C^{\ast }$-correspondence of $C^{\ast }$-dynamical systems unitarily uniquely dilates under the KSGNS construction to an equivariant $C^{\ast }$-correspondence of the dynamical systems.

v3: fixed typos, generalized the construction of the categories, and fixed arguments in section 5 regarding continuity of the functors and composition map