Connes' Embedding Problem and Lance's WEP

TL;DR

This paper proves that a II_1 factor embeds into the ultraproduct of the hyperfinite II_1 factor if and only if it satisfies the W*-analogue of Lance's weak expectation property, establishing a key equivalence in operator algebra theory.

Contribution

It provides a self-contained proof of the equivalence between embeddability into the hyperfinite II_1 ultraproduct and satisfying the W*-weak expectation property.

Findings

Establishes the equivalence between embeddability and W*-WEP for II_1 factors

Provides a new self-contained proof of this fundamental result

Clarifies the relationship between Connes' embedding problem and Lance's WEP

Abstract

A II_1 factor embeds into the ultraproduct of the hyperfinite II_1 factor if and only if it satisfies the W*-analogue of Lance's weak expectation property. This note gives a self contained proof of this fact.