Actions of compact and discrete quantum groups on operator systems

TL;DR

This paper introduces the concept of quantum group actions on operator systems, explores their injectivity properties, and establishes duality results relating equivariant and dual injectivity, with applications to crossed products.

Contribution

It defines quantum group actions on operator systems, studies their injectivity, and proves a duality theorem linking equivariant and dual injectivity in this context.

Findings

Established a duality between equivariant and dual injectivity.

Provided a description of the equivariant injective envelope of crossed products.

Introduced a framework for quantum group actions on operator systems.

Abstract

We introduce the notion of an action of a discrete or compact quantum group on an operator system, and study equivariant operator system injectivity. We then prove a duality result that relates equivariant injectivity with dual injectivity on associated crossed products. As an application, we give a description of the equivariant injective envelope of the reduced crossed product built from an action of a discrete quantum group on an operator system.