The theory of exact -algebras was introduced by Kirchberg and has been influential in recent development of -algebras. A fundamental result on exact -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.
Narutaka OZAWA. "Weakly exact von Neumann algebras." J. Math. Soc. Japan 59 (4) 985 - 991, October, 2007. https://doi.org/10.2969/jmsj/05940985