A Kirchberg-type tensor theorem for operator systems

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 $$\mathfrak C_I \otimes_{\min} B(H) = \mathfrak C_I \otimes_{\max} B(H).$$ 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 $$\mathfrak C_I \otimes_{\min} \mathfrak C_I =\mathfrak C_I \otimes_{\rm c} \mathfrak C_I.$$ 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.