Functoriality of the KSGNS Construction for Intertwiners of Strict Positive $C^*$-Correspondences

TL;DR

This paper demonstrates that the KSGNS construction acts as a functor on a category of positive $C^*$-correspondences, enabling a functorial view and showing unique dilations of equivariant correspondences.

Contribution

It introduces a functorial perspective of the KSGNS construction for positive $C^*$-correspondences and establishes unique dilations for equivariant correspondences in $C^*$-dynamical systems.

Findings

KSGNS construction is a functor on a category of positive $C^*$-correspondences.

Every strict positive equivariant $C^*$-correspondence unitarily dilates under KSGNS.

Provides a functorial framework for equivariant $C^*$-correspondences in dynamical systems.

Abstract

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