On a Characterization of the Weak Expectation Property (WEP)

TL;DR

This paper provides a detailed proof of a new characterization of the Weak Expectation Property (WEP), linking it to the Connes embedding problem and addressing a longstanding conjecture in operator algebras.

Contribution

It offers the first comprehensive proof of a new WEP characterization, advancing understanding of its connection to the Connes embedding problem.

Findings

New characterization of WEP established

Connection between WEP and Connes embedding problem clarified

Provides foundational insights for future research in operator algebras

Abstract

We give a detailed proof of a new characterization of the Weak Expectation Property (WEP) announced by Haagerup in the 1990's but unavailable (in any form) till now. Our main result is motivated by a well known conjecture of Kirchberg, which is equivalent to the Connes embedding problem. We review the basic relevant facts connecting our main theorem with the latter conjecture, along the lines of our forthcoming lecture notes volume on the Connes-Kirchberg problem.