Quotients, exactness, and nuclearity in the operator system category

Abstract

We continue our study of tensor products in the operator system category. We define operator system quotients and exactness in this setting and refine the notion of nuclearity by studying operator systems that preserve various pairs of tensor products. One of our main goals is to relate these refinements of nuclearity to the Kirchberg conjecture. In particular, we prove that the Kirchberg conjecture is equivalent to the statement that every operator system that is (min,er)-nuclear is also (el,c)-nuclear. We show that operator system quotients are not always equal to the corresponding operator space quotients and then study exactness of various operator system tensor products for the operator system quotient. We prove that an operator system is exact for the min tensor product if and only if it is (min,el)nuclear. We give many characterizations of operator systems that are (min,er)-nulcear, (el,c)nuclear, (min,el)-nulcear and (el,max)-nuclear. These characterizations involve operator system analogues of various properties from the theory of C*-algebras and operator spaces, including the WEP and LLP.

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.…”

The Functional Analysis of Quantum Information Theory

“…Analogous to the notion of exactness for C * -algebras and operator systems, there is a notion of exactness for operator systems as well -see [KPTT2]. For an operator system S, we consider its Banach space dual S * and endow it with a matrix ordering as we did for S d above.…”

Lovász theta type norms and operator systems

“…In [7], it was shown that the vector space quotient S/J can be turned into an operator system, called the quotient operator system as follows. Let D n (S/J ) be the set of all (x i,j + J ) ∈ M n (S/J ) for which there exists (y i,j ) ∈ M n (J ) such that (x i,j + y i,j ) ∈ M n (S) + .…”

“…Following [7], given X ∈ M n (S) so thatẊ ∈ M n (S/J ) we let Ẋ osp (resp. Ẋ osy ) denote the operator space (resp.…”