Permanence properties of $C^{\ast }$-exact groups

Abstract

It is shown that the class of exact groups, as defined in a previous paper, is closed under various operations, such as passing to a closed subgroup and taking extensions. Taken together, these results imply, in particular, that all almost-connected locally compact groups are exact. The proofs of the permanence properties use a result relating the exactness of sequences of maps in which corresponding algebras are strongly Morita equivalent. The statement of this result is based on a notion of reduced twisted crossed product for covariant systems which are twisted in the sense of Green. The theory of these reduced twisted crossed products and the proof of the exactness result are given in the first part of the paper.