Group homomorphisms induced by isometries

TL;DR

This paper characterizes isometries between spaces of almost periodic functions on locally compact groups, revealing their structure as induced by group homomorphisms and characters under certain conditions.

Contribution

It establishes new structural theorems describing how isometries of almost periodic function spaces correspond to group homomorphisms and characters, extending previous understanding.

Findings

Isometries induce group homomorphisms and characters.

Results apply to $\sigma$-compact maximally almost periodic groups.

Connected Abelian groups have isometries preserving trigonometric polynomials.

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 $(Tf)(h)=a\cdot \alpha (h)~\cdot f(t(h-h_0))\text{ for all }h\in H_0\text{ and for all }f\in C(G)\text{.}$