On a Notion of Exactness for Reduced Free Products of C*-Algebras

TL;DR

This paper explores a modified notion of exactness for C*-algebras using reduced free products, demonstrating preservation of exact sequences and behavior under ultrapowers, thus extending the understanding of algebraic exactness.

Contribution

It introduces a new perspective on exactness for C*-algebras via reduced free products and shows how this notion preserves exact sequences and interacts with ultrapowers.

Findings

Reduced free product preserves exact sequences of C*-algebras.

Adjoining operators freely behaves well with ultrapowers.

Modified exactness notion extends classical properties.

Abstract

We will study some modifications to the notion of an exact C*-algebra by replacing the minimal tensor product with the reduced free product. First we will demonstrate how the reduced free product of a short exact sequence of C*-algebras with another C*-algebra may be taken. It will then be demonstrated that this operation preserves exact sequences. We will also establish that adjoining arbitrary k-tuples of operators in a free way behaves well with respect to taking ultrapowers.