Exactness vs C*-exactness for certain non-discrete groups
It is known that exactness for a discrete group $G$ is equivalent to C*-exactness, i.e., the exactness of the 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.