Tensor Products of Operator Systems

2009/10/12 by Ali S. Kavruk, Kavruk, Ali S., Vern I. Paulsen +5 · 9 citations
Mathematics · Physics and Astronomy · #46L05 #47L25 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA) #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.0910.2210

openalex publication_date 2009/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Completely positive linear maps on complex matrices

Cited by

• Matrix-Test Duality: A Support-Function Characterization for C^*-Convex Families of CP Maps
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• Soft inductive limits of operator systems and a noncommutative Lazar-Lindenstrauss theorem
• Boundary representations, geometry of matrix ranges, and C^*-envelopes of finite-dimensional operator systems
• Real decomposable maps on operator systems
• Cantor correlations I. Operator systems and Cantor games
• Unitary induced channels and Tsirelson's problem
• Four-dimensional operator systems without the lifting property
• Local presentability and monadicity of forgetful functors for operator algebraic categories

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