(CA) Topological Groups

Abstract

A locally compact topological group G is called (C4) if the group of inner automorphisms of G is closed in the group of all bicontinuous automorphisms of G. We show that each non-(C4) locally compact connect- ed group G can be written as a semidirect product of a (C A) locally compact connected group by a vector group. This decomposition yields a natural dense imbedding of G into a iC A) locally compact connected group P, such that each bicontinuous automorphism of G can be extended to a bicontin- uous automorphism of P. This imbedding and extension property enables us to derive a sufficient condition for the normal part of a semidirect product decomposition of a (C A) locally compact connected group to be (C A). 1. Introduction. The purpose of this paper is to extend the results in Zerling (5) to the case of locally compact connected groups. If G and H are topological groups and A(G) defined by p$(h)(g) = y~x(h<p(g)h~x) is a continuous homomorphism (2). If <p is a closed imbedding, such that <p(G) is normal in TT, then pG is clearly continuous. For any locally compact group TT we let IH(h) denote the inner automor- phism of H determined by h G TT. More generally, if A is a subset of TT, Ih(A) will denote the set of all inner automorphisms of TT determined by the elements of A. T#(TT) will be written as T(TT), and the continuous homomor- phism h i-» IHih) of TT onto T(TT) will be denoted by IH. If A is a locally compact connected group and t/<: TT —* AiN) is a contin- uous homomorphism of some connected topological group TT into AiN), then N © H will denote the semidirect product of N by TT, which is determined by \p. On the other hand, if G is a locally compact connected group containing a closed normal connected subgroup N and a closed connected subgroup TT, such that G = NH, N n TT = {e}, and such that the restriction of pN to TT is one-to-one, then whenever we write pNiH) it will be understood that the