$C^*$-Operator systems and crossed products

TL;DR

This paper develops a unified framework for defining and analyzing universal and reduced crossed products of operator systems and spaces by group actions, extending classical duality theorems to these broader contexts.

Contribution

It introduces the concepts of $C^*$-operator systems and bimodules, enabling consistent constructions of crossed products and extending duality theorems beyond $C^*$-algebras.

Findings

Universal properties of crossed products are fully characterized.

Classical duality theorems extend to $C^*$-operator systems and bimodules.

Framework accommodates non-discrete groups.

Abstract

The purpose of this paper is to introduce a consistent notion of universal and reduced crossed products by actions and coactions of groups on operator systems and operator spaces. In particular we shall put emphasis to reveal the full power of the universal properties of the the universal crossed products. It turns out that to make things consistent, it seems useful to perform our constructions on some bigger categories which allow the right framework for studying the universal properties and which are stable under the construction of crossed products even for non-discrete groups. In the case of operator systems, this larger category is what we call a $C^*$-operator system, i.e., a selfadjoint subspace $X$ of some $\mathcal B(H)$ which contains a $C^*$-algebra $A$ such that $AX=X=XA$. In the case of operator spaces, the larger category is given by what we call $C^*$-operator bimodules. After we introduced the respective crossed products we show that the classical Imai-Takai and Katayama duality theorems for crossed products by group (co-)actions on $C^*$-algebras extend one-to-one to our notion of crossed products by $C^*$-operator systems and $C^*$-operator bimodules.