Group C*-algebras as decreasing intersection of nuclear C*-algebras

TL;DR

This paper demonstrates that for exact discrete groups, their reduced group C*-algebras can be expressed as decreasing intersections of nuclear C*-algebras, specifically as intersections of isomorphic copies of the Cuntz algebra , with implications for extensions of free group C*-algebras.

Contribution

It establishes a new structural characterization of reduced group C*-algebras as decreasing intersections of nuclear C*-algebras, including those isomorphic to , and explores their extensions.

Findings

Reduced group C*-algebras are intersections of decreasing sequences of nuclear C*-algebras.

When  has the AP, the reduced group C*-algebra is realized as such an intersection.

Extensions of reduced free group C*-algebras can also be realized as intersections of nuclear C*-algebras.

Abstract

We prove that for every exact discrete group $\Gamma$, there is an intermediate C*-algebra between the reduced group C*-algebra and the intersection of the group von Neumann algebra and the uniform Roe algebra which is realized as the intersection of a decreasing sequence of isomorphs of the Cuntz algebra $\mathcal{O}_2$. In particular, when $\Gamma$ has the AP (approximation property), the reduced group C*-algebra is realized in this way. We also study extensions of the reduced free group C*-algebras and show that any exact absorbing or unital absorbing extension of it by any stable separable nuclear C*-algebra is realized in this way.