The Weak Expectation Property and Riesz Interpolation

Abstract

We show that Lance's weak expectation property is connected to tight Riesz interpolations in lattice theory. More precisely we first prove that if A \subset B(H) is a unital C*-subalgebra, where B(H) is the bounded linear operators on a Hilbert space H, then A has (2,2) tight Riesz interpolation property in B(H) (defined below). An extension of this requires an additional assumption on A: A has (2,3) tight Riesz interpolation property in B(H) at every matricial level if and only if A has the weak expectation property. Let $J = span{(1,1,-1,-1,-1)}$ in $C^5$ . We show that a unital C*-algebra A has the weak expectation property if and only if $A \otimesmin (C^5/J) = A \otimesmax (C^5/J)$ (here \otimesmin and \otimesmax are the minimal and the maximal operator system tensor products, respectively, and $C^5/J$ is the operator system quotient of $C^5$ by $J$). We express the Kirchberg conjecture (KC) in terms of a four dimensional operator system problem. We prove that KC has an affirmative answer if and only if $C^5/J$ has the double commutant expectation property if and only if $C5/J \otimesmin C5/J = C5/J \otimesc C5/J$ (here \otimesc represents the commuting operator system tensor product).