Tensor products of operator systems

By A. Kavruk, Vern I. Paulsen, Ivan G. Todorov, M. Tomforde. 2009

151 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a755a51fe3e674f29c260e201c847e7726efd036

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In Algebra and Number Theory

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Characterisations of the weak expectation property (2013, methodology): “For a graph G on n vertices, let SG be the “graph operator system” (see [16]) SG = span{Ei,j, Ek,k : k = 1, .”
Operator system quotients of matrix algebras and their tensor products (2011, methodology): “Tensor Products, Quotients, and Duals of Operator Systems: A Review 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.”
Nuclearity Related Properties in Operator Systems (2011, methodology): “In this section we recall the axiomatic definition of tensor products in the category of operator systems and review properties of several tensor products established in [22].”
Tensor Products of C*-Algebras and Operator Spaces (2020, background): “Curiously, operator systems made a reappearance more recently and their tensor products were investigated thoroughly in a series of papers, notably [150, 151].”
Connes' embedding problem and Tsirelson's problem (2010, methodology): “We must note that the notation ⊗cmax here is different from the one used in [15] (⊗c).”
C*-algebras with the weak expectation property and a multivariable analogue of Ando's theorem on the numerical radius (2011, background): “The fundamental results for a theory of operator system tensor products are developed in [8, 9].”
Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture (2013, background): “Other very recent discoveries include the fact that Connes’ embedding conjecture is related to Tsirelson’s problem, a major open problem in Quantum Information Theory (see [49], [66], and [93]) and that it is connected to the “minimal” and “commuting” tensor products [73] of some group operator syst”
QUANTUM CHROMATIC NUMBERS VIA OPERATOR SYSTEMS (2013, methodology): “Throughout the paper, we will use notions and results about operator systems and their tensor products which can be found in [15], [14] and [16].”
Quotients, exactness, and nuclearity in the operator system category (2010, background): “ the equality of certain tensor products. Namely, a C*-algebra Ahas the WEP if and only if A⊗ max B⊆I(A) ⊗ max B, for every C*-algebra B(here I(A) is the injective envelope of A). In our earlier work [9], we showed that the maximal tensor product on the category of C*-algebras has two distinct natur”