Quotients, exactness, and nuclearity in the operator system category

Ali Kavruk, Vern Paulsen, Ivan Todorov, Mark Tomforde

Research output: Contribution to journalArticlepeer-review

36 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)321-360
Number of pages40
JournalAdvances in Mathematics
Volume235
DOIs
Publication statusPublished - 2013

ASJC Scopus subject areas

  • Mathematics(all)

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