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.
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.
Some recent research on the tensor products of operator systems and ensuing nuclearity properties in this setting raised many stability problems. In this paper we examine the preservation of these nuclearity properties including exactness, local liftability and the double commutant expectation property under basic algebraic operations such as quotient, duality, coproducts and tensorial products. We show that, in the finite dimensional case, exactness and the lifting property are dual pairs, that is, an operator system S is exact if and only if the dual operator system S d has the lifting property. Moreover, the lifting property is preserved under quotients by null subspaces.Again in the finite dimensional case we prove that every operator system has the k-lifting property in the sense that whenever ϕ : S → A/I is a unital and completely positive map, where A is a C*-algebra and I is an ideal, then ϕ possess a unital k-positive lift on A, for every k. This property provides a novel proof of a classical result of Smith and Ward on the preservation of matricial numerical ranges of an operator.The Kirchberg conjecture naturally falls into this context. We show that the Kirchberg conjecture is equivalent to the statement that the five dimensional universal operator system generated by two contraction (S2) has the double commutant expectation property. In addition to this we give several equivalent statements to this conjecture regarding the preservation of various nuclearity properties under basic algebraic operations.We show that the Smith Ward problem is equivalent to the statement that every three dimensional operator system has the lifting property (or exactness). If we suppose that both the Kirchberg conjecture and the Smith Ward problem have an affirmative answer then this implies that every three dimensional operator system is C*-nuclear. We see that this property, even under most favorable conditions, seems to be hard to verify.
We formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of the free group Fn on n generators, as well as the operator systems of the free products of finitely many copies of the two-element group Z 2 . We examine various tensor products of group operator systems, including the minimal, the maximal, and the commuting tensor products. We introduce a new tensor product in the category of operator systems and formulate necessary and sufficient conditions for its equality to the commuting tensor product in the case of group operator systems. We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlation boxes.2010 Mathematics Subject Classification. Primary 46L06, 46L07; Secondary 46L05, 47L25, 47L90.
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