Quotients, exactness, and nuclearity in the operator system category

By A. Kavruk, Vern I. Paulsen, Ivan G. Todorov, M. Tomforde. 2010

127 citations recorded by the source corpus.

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)-nuclear, (el,c)-nuclear, (min,el)-nuclear 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.

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

View Paper: https://www.semanticscholar.org/paper/56c44b662ba255709c7a964c83220a6f558f37e1

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In Algebra and Number Theory

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Citations in the corpus

QUANTUM CHROMATIC NUMBERS VIA OPERATOR SYSTEMS (2013, methodology): “and ﬁts with Tsirelson’s work on non-signalling boxes [21]. 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]. We refer to these papers about basic properties of the tensor products QUANTUM CHROMATIC NUMBE”
Nuclearity Related Properties in Operator Systems (2011, background): “r systems are also used in quantum mechanics ([38], e.g.). However a systematic study of tensor products on this category along with the characterization of nuclearity properties waited till [22] and [21] (see also [17]). This series of papers raised several questions; namely, the stability of these”