Homomorphisms on groups and induced maps on certain algebras of measures

Abstract

Suppose that φ \varphi is a continuous homomorphism of a locally compact group G G into another such group, H H , then φ \varphi induces in a natural way a homomorphism φ ∗ {\varphi ^ \ast } of the measure algebra of G G , called M ( G ) M(G) , into M ( H ) M(H) . The action of φ ∗ {\varphi ^ \ast } on the subspace M 0 ( G ) {M_0}(G) is studied in this paper. The space M 0 ( G ) {M_0}(G) is the nonabelian analogue to the space of measures on a locally compact abelian group whose Fourier-Stieltjes transforms vanish at infinity, and is defined herein. We prove that if φ \varphi is an open homomorphism then φ ∗ ( M 0 ( G ) ) ⊂ M 0 ( H ) {\varphi ^ \ast }({M_0}(G)) \subset {M_0}(H) . If G G and H H are abelian and φ \varphi is not open, then φ ∗ ( M ( G ) ) ∩ M 0 ( H ) = { 0 } {\varphi ^ \ast }(M(G)) \cap {M_0}(H) = \{ 0\} . The main tool for this theorem is the fact, proved herein, that φ \varphi is open if and only if its adjoint, φ ^ : H ^ → G ^ \hat \varphi :\hat H \to \hat G , is proper (where G ^ , H ^ \hat G,\hat H are the character groups of G , H G,H resp.). Further properties of M 0 ( G ) {M_0}(G) for abelian or compact groups G G are derived.