Tensor Products of Operator Systems

TL;DR

This paper develops a systematic framework for tensor products of operator systems, exploring their properties, examples, and nuclearity, and establishing connections with C*-algebra tensor products.

Contribution

It provides an axiomatic foundation for tensor products of operator systems, characterizes various types, and relates them to classical C*-algebra tensor products.

Findings

Characterization of minimal and maximal tensor products

Identification of tensor product properties in operator systems

Existence of an operator system with unique tensor product behavior

Abstract

The purpose of the present paper is to lay the foundations for a systematic study of tensor products of operator systems. After giving an axiomatic definition of tensor products in this category, we examine in detail several particular examples of tensor products, including a minimal, maximal, maximal commuting, maximal injective and some asymmetric tensor products. We characterize these tensor products in terms of their universal properties and give descriptions of their positive cones. We also characterize the corresponding tensor products of operator spaces induced by a certain canonical inclusion of an operator space into an operator system. We examine notions of nuclearity for our tensor products which, on the category of C*-algebras, reduce to the classical notion. We exhibit an operator system S which is not completely order isomorphic to a C*-algebra yet has the property that for every C*-algebra A, the minimal and maximal tensor product of S and A are equal.