  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.
