Crossed products of operator systems

TL;DR

This paper develops a new framework for crossed products of operator systems under group actions, analyzing their properties and relationships with $C^*$-algebras, and provides a negative answer to a specific open question in the field.

Contribution

It introduces three canonical crossed product constructions for operator systems with group actions and studies their properties and implications.

Findings

Hyperrigidity is preserved under two of the three crossed products.

The behavior of crossed products under $G$-equivariant maps and tensor products is characterized.

A negative answer is given to a question about operator algebra crossed products posed by Katsoulis and Ramsey.

Abstract

In this paper we introduce the crossed product construction for a discrete group action on an operator system. In analogy to the work of E. Katsoulis and C. Ramsey, we describe three canonical crossed products arising from such a dynamical system. We describe how these crossed product constructions behave under $G$-equivariant maps, tensor products, and the canonical $C^*$-covers. We show that hyperrigidity is preserved under two of the three crossed products. Finally, using A. Kavruk's notion of an operator system that detects $C^*$-nuclearity, we give a negative answer to a question on operator algebra crossed products posed by Katsoulis and Ramsey.