Weak exactness for C⁎-algebras and application to condition (AO)

Kirchberg introduced weak exactness for von Neumann algebras as an analogue of exactness for C⁎-algebras. Ozawa found useful characterizations of weak exactness and he proved that weak exactness for group von Neumann algebras is equivalent to exactness of the groups. We generalize this concept to inclusions of C⁎-algebras in von Neumann algebras and study some fundamental properties and relationship to Kirchbergʼs weak exactness. We then give some permanence properties which are similar to those of exact groups. In the last section, we study a similar condition to Ozawaʼs condition (AO) with our weak exactness and generalize Ozawaʼs theorem for bi-exact groups. As a corollary, we give new examples of prime factors.