Weakly exact von Neumann algebras

Abstract

The theory of exact $C^{*}$-algebras was introduced by Kirchberg and has been influential in recent development of $C^{*}$-algebras. A fundamental result on exact $C^{*}$-algebras is a local characterization of exactness. The notion of weakly exact von Neumann algebras was also introduced by Kirchberg. In this paper, we give a local characterization of weak exactness. As a corollary, we prove that a discrete group is exact if and only if its group von Neumann algebra is weakly exact. The proof naturally involves the operator space duality.