An intrinsic order-theoretic characterization of the weak expectation property

TL;DR

This paper characterizes the weak expectation property of operator systems using an order-theoretic approach, linking it to Wittstock's matricial Riesz separation property, and provides a noncommutative analog of classical results.

Contribution

It offers a new intrinsic order-theoretic characterization of the weak expectation property for operator systems, extending classical Riesz separation concepts to the noncommutative setting.

Findings

Operator systems satisfy the weak expectation property iff their matrix amplifications satisfy the matricial Riesz separation property.

Provides a noncommutative analog of classical Riesz separation characterization for simplex spaces.

Establishes a link between operator system properties and order-theoretic separation principles.

Abstract

We prove the following characterization of the weak expectation property for operator systems in terms of Wittstock's matricial Riesz separation property: an operator system $S$ satisfies the weak expectation property if and only if $M_{q}(S)$ satisfies the matricial Riesz separation property for every $q\in \mathbb{N}$. This can be seen as the noncommutative analog of the characterization of simplex spaces among function systems in terms of the classical Riesz separation property.