Operator system quotients of matrix algebras and their tensor products

By Douglas R. Farenick, Vern I. Paulsen. 2011

103 citations recorded by the source corpus.

An operator system modulo the kernel of a completely positive linear map of the operator system gives rise to an operator system quotient. In this paper, operator system quotients and quotient maps of certain matrix algebras are considered. Some applications to operator algebra theory are given, including a new proof of Kirchberg's theorem on the tensor product of B(H) with the group C*-algebra of a countable free group. We also show that an affirmative solution to the Connes Embedding Problem is implied by various matrix-theoretic problems, and we give a new characterisation of unital C*-algebras that have the weak expectation property.

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

View Paper: https://www.semanticscholar.org/paper/aa0db33a8627ebd6d0e05c5a5606361627dda094

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In Algebra and Number Theory

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In Statistical and Nonlinear Physics

Citations in the corpus

Characterisations of the weak expectation property (2013, background): “(We note that the map sending Ei,j to uiu ∗ j was considered in [10] but since {u1, .”
About the Connes Embedding Conjecture---Algebraic approaches--- (2012, background): “ product satisfy the axiom of operator systems, and that the formal identity map from S ⊗max T to S ⊗min T is c.p. These two tensor products are dual to each other in the following sense. Theorem 35 ([FP]). Let S and T be ﬁnite-dimensional operator systems. Then the natural isomorphism (S ⊗min T )d ”
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”
Tensor Products of C*-Algebras and Operator Spaces (2020, methodology): “This led to several characterizations of the WEP (see [90, 149, 153, 152, 91, 92]), connected to the Connes-Kirchberg problem, but for lack of space (and energy) we chose not to cover this.”
Nuclearity Related Properties in Operator Systems (2011, background): “qd(δ 1) = δ 2 requires that Jis in the kernel of δ 2 and consequently it has to be a null subspace. The converse of the above result is also true which is referred as the First Isomorphism Theorem in [10]. For completeness of the paper we include the proof. Theorem 2.8 (Farenick, Paulsen). Let Sbe a”