Homomorphisms on Groups and Induced Maps on Certain Algebras of Measuresc)

Suppose that y is a continuous homomorphism of a locally compact group G into another such group, H, then ¡p induces in a natural way a homomorphism 9>* of the measure algebra of G, called M(G), into M(H). The action of G (where G is the character group of G), carried out in §1, gives a converse to the above theorem; in fact, if cp is not open, then <p*(M(G)) C\ Mo(77) = {0}. The main tool for this theorem is the fact, proved in §1, that 9 is open if and only if 9 is proper (i.e., the inverse image under 9 of a compact set is compact). In §3 further properties of M0iG) for abelian or compact groups G are derived. One result is an extension of a classical result of Rajchman to abelian groups with a certain property, and examples are presented to illustrate this property. Another result says that if <p is an ergodic homomorphism of a compact Received by the editors January 23, 1970 and, in revised form, June 6, 1970. AMS 1968 subject classifications. Primary 2265, 4250, 4256, 4680.