# Exactness vs C*-exactness for certain non-discrete groups

## TL;DR

This paper investigates the relationship between exactness and C*-exactness in locally compact groups, proving their equivalence for certain classes including those with tracial states and specific totally disconnected groups.

## Contribution

It establishes the equivalence for a new class of groups, extending previous results and providing original proofs without relying on inner amenability.

## Key findings

- Equivalence holds for groups with reduced C*-algebra admitting a tracial state.
- Totally disconnected locally compact IN groups satisfy the equivalence.
- A class of groups introduced by Suzuki also satisfies the equivalence.

## Abstract

It is known that exactness for a discrete group is equivalent to C*-exactness, i.e., the exactness of its reduced C*-algebra. The problem of whether this equivalence holds for general locally compact groups has recently been reduced by Cave and Zacharias to the case of totally disconnected unimodular groups. We prove that the equivalence does hold for a class of groups that includes all locally compact groups with reduced C*-algebra admitting a tracial state. As a consequence, we present original proofs that totally disconnected locally compact (tdlc) invariant-neighbourhood (IN) groups and a class of groups introduced by Suzuki satisfy the equivalence problem, without using inner amenability.