Homomorphisms on groups and induced maps on certain algebras of measures

Abstract:

Suppose that $\varphi$ is a continuous homomorphism of a locally compact group $G$ into another such group, $H$, then $\varphi$ induces in a natural way a homomorphism ${\varphi ^ \ast }$ of the measure algebra of $G$, called $M(G)$, into $M(H)$. The action of ${\varphi ^ \ast }$ on the subspace ${M_0}(G)$ is studied in this paper. The space ${M_0}(G)$ 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 $\varphi$ is an open homomorphism then ${\varphi ^ \ast }({M_0}(G)) \subset {M_0}(H)$. If $G$ and $H$ are abelian and $\varphi$ is not open, then ${\varphi ^ \ast }(M(G)) \cap {M_0}(H) = \{ 0\}$. The main tool for this theorem is the fact, proved herein, that $\varphi$ is open if and only if its adjoint, $\hat \varphi :\hat H \to \hat G$, is proper (where $\hat G,\hat H$ are the character groups of $G,H$ resp.). Further properties of ${M_0}(G)$ for abelian or compact groups $G$ are derived.