A Kirchberg-type tensor theorem for operator systems

Kyung Hoon Han

Vol. 286 (2017), No. 1, 91–124

DOI: 10.2140/pjm.2017.286.91

Abstract

We construct operator systems ${\mathfrak{ℭ}}_I$ that are universal in the sense that all operator systems can be realized as their quotients. They satisfy the operator system lifting property. Without relying on the theorem by Kirchberg, we prove the Kirchberg-type tensor theorem

$${\mathfrak{ℭ}}_I{\otimes }_{min}B\left(H\right)={\mathfrak{ℭ}}_I{\otimes }_{max}B\left(H\right).$$

Combining this with a result of Kavruk, we give a new operator system theoretic proof of Kirchberg’s theorem and show that Kirchberg’s conjecture is equivalent to its operator system analogue

$${\mathfrak{ℭ}}_I{\otimes }_{min}{\mathfrak{ℭ}}_I={\mathfrak{ℭ}}_I{\otimes }_c{\mathfrak{ℭ}}_I.$$

It is natural to ask whether the universal operator systems ${\mathfrak{ℭ}}_I$ are projective objects in the category of operator systems. We show that an operator system from which all unital completely positive maps into operator system quotients can be lifted is necessarily one-dimensional. Moreover, a finite-dimensional operator system satisfying a perturbed lifting property can be represented as the direct sum of matrix algebras. We give an operator system theoretic approach to the Effros–Haagerup lifting theorem.

Keywords

operator system, tensor product, quotient, Kirchberg's theorem

Mathematical Subject Classification 2010

Primary: 46L06, 46L07, 47L07

Milestones

Received: 17 July 2015

Accepted: 10 July 2016

Published: 9 December 2016