The Weak Expectation Property and Riesz Interpolation

TL;DR

This paper links the weak expectation property of C*-algebras to Riesz interpolation in lattice theory, providing new characterizations and connecting to the Kirchberg conjecture through operator system tensor products.

Contribution

It establishes a novel connection between the weak expectation property and Riesz interpolation, and reformulates the Kirchberg conjecture in terms of operator system tensor products.

Findings

A C*-algebra has the weak expectation property iff certain tensor products coincide.

Unital C*-subalgebras have (2,2) tight Riesz interpolation in B(H).

KC is equivalent to properties of the operator system C^5/J.

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).