Group homomorphisms induced by isometries

Abstract

Let $G$ and $H$ be locally compact groups and consider their associate spaces of almost periodic functions $AP(G)$ and $AP(H)$. We investigate the continuous group homomorphisms induced by isometries of $AP(G)$ into $AP(H)$. Among others, the following results are proved: {\bf Theorem} Let $G$ and $H$ be $\sigma$-compact maximally almost periodic locally compact groups. Suppose that $T$ is a non-vanishing linear isometry of $AP(G)$ into $AP(H)$ that respects finite dimensional unitary representations. Then there is a closed subgroup $H_0\subseteq H$, a continuous group homomorphism $t$ of $H_0$ onto $G$ and an character $\gamma\in \widehat{H}$ such that $(Tf)(h)=\gamma (h)~f(t(h))$ for all $h\in H_0$ and for all $f\in C(G)$. {\bf Theorem} Let $G$ and $H$ be $LC$ Abelian groups and $H$ is connected. Suppose that $T$ is a non-vanishing linear isometry of $AP(G)$ into $AP(H)$ that preserves trigonometric polynomials. Then there is a closed subgroup $H_0\subseteq H$, a continuous group homomorphism $t$ of $H_0$ onto $G$, an element $h_0\in H_0$, a character $\alpha \in \widehat{H}$ and an unimodular complex number $a$ such that $(Tf)(h)=a\cdot \alpha (h)~\cdot f(t(h-h_0))\text{ for all }h\in H_0\text{ and for all }f\in C(G)\text{.}$