Weak Exactness for C*-algebras and Application to Condition (AO)

TL;DR

This paper extends the concept of weak exactness from C*-algebras to their inclusions in von Neumann algebras, explores related properties, and applies these ideas to generalize Ozawa's condition (AO), leading to new prime factor examples.

Contribution

It introduces a generalized notion of weak exactness for inclusions of C*-algebras in von Neumann algebras and applies it to extend Ozawa's theorem for bi-exact groups.

Findings

Generalization of weak exactness to inclusions of C*-algebras in von Neumann algebras

Characterizations and permanence properties similar to exact groups

New examples of prime factors derived from the generalized framework

Abstract

We generalize Kirchberg's weak exactness to inclusions of C*-algebras in von Neumann algebras and study some characterizations and permanence properties which are similar to those of exact groups. We then consider a similar condition to Ozawa's condition (AO) with our weak exactness and generalize Ozawa's theorem for bi-exact groups. As a corollary, we give new examples of prime factors.