On Extensions of Locally Compact Groups

Abstract

Let Q be a locally compact group, and let K be a Q-module, that is, a locally compact Abelian group on which Q operates continuously as a group of automorphisms. An extension of Q by K is a locally compact group, G, together with a homomorphism, p, of G onto Q, and a topological isomorphism, i, of K onto the kernel of p, such that p induces a topological isomorphism of G/i (K) onto Q in such a way that the natural action of Q G/i (K) on K i(K) by inner automorphisms is the given action of Q on K. (See [8] for an extensive bibliography). Mackey [6] showed that if K and Q are separable, in the sense of satisfying the second axiom of countability, then to each extension, G, of Q by K there corresponds a (non-unique) normalized Borel 2-cocycle on Q with values in K, that is, a Borel function c from Q X Q into K satisfying the cocycle identity