Tensor products of operator systems

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.

Various notions of positivity for bi-linear maps and applications to tri-partite entanglement

“…We proceed to recall the definition [17] of the maximal tensor product of operator systems. For two operator systems S and T , the sets…”

“…It was shown in [17] that the linearization of a bi-linear map φ : S ×T → R between operator systems is completely positive with respect to the operator system maximal tensor products if and only if the following condition In fact, there is no way to control the matrix size over the range spaces with the above condition (1). This motivates the following definition.…”

Operator system quotients of matrix algebras and their tensor products

“…In this section we review some of the fundamental facts, established in [10], [9], concerning tensor products, quotients, and duals of operator systems, and introduce the notion of a complete quotient map. Some basic notation: (i) the Archimedean order unit e of an operator system S is generally denoted by 1, but we will sometimes revert to the use of e in cases where the order unit is not canonically given (for example, when considering duals of operator systems); (ii) for a linear map φ : S → T , the map φ (n) : M n (S ) → M n (T ) is defined by φ (n) ([x ij ] i,j ) = [φ(x ij )] i,j ; (iii) for any operator systems S and T , S ⊗ T shall denote their algebraic tensor product.…”

“…Note that σ is well defined. Furthermore, σ is jointly completely positive and "unital", and so by the universal property of the max tensor product [10,Theorem 5.8], there is a ucp extension σ :…”

Tensor products of function systems revisited

“…Recently, Paulsen and Tomforde introduced the Archimedeanization [PT] and this has been applied to operator system theory in a series of papers [PTT,KPTT1,KPTT2]. In this paper, we focus on its application to function systems.…”