Co-amenability and Connes’s embedding problem

Abstract

A longstanding open question of Connes asks whether any finite von Neumann algebra embeds into an ultraproduct of finite-dimensional matrix algebras. As of yet, algebras verified to satisfy the Connes’s embedding property belong to just a few special classes (e.g., amenable algebras and free group factors). In this article, we prove that von Neumann algebras satisfying Popa’s co-amenability have Connes’s embedding property.