Group homomorphisms induced by isometries
Salvador Hernández
TL;DR
The paper addresses when isometries between spaces of almost periodic functions $AP(G)$ and $AP(H)$ on maximally almost periodic locally compact groups induce continuous group homomorphisms. It combines Holsztyński's canonical form with Tannaka–Kreĭn duality and Bohr compactification to connect functional-analytic isometries with underlying group structure, first for compact groups and then extended to σ-compact MAP locally compact groups. The main results show that non-vanishing linear isometries that respect finite-dimensional unitary representations must be, on a closed subgroup $H_0$, of the form $(Tf)(h)=\, ext{γ}(h)\,f(t(h))$ with a continuous homomorphism $t:H_0 o G$ and a character γ, with refinements when trigonometric polynomials are preserved and $H$ is connected. When extended to MAP groups via $AP(G)$ and Bohr compactifications, these isometries typically force a Bohr-continuous (and often globally topological) homomorphism $t:H o G$, yielding $(Tf)(h)= ext{γ}(h) olinebreak[4] f(t(h))$ on all of $H$ in favorable cases. The work contributes a variant of a duality theory for MAP groups and clarifies the rigidity of isometries in this harmonic-analysis setting, including explicit counterexamples demonstrating the necessity of non-vanishing assumptions.
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 $σ$-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 $γ\in \widehat{H}$ such that $(Tf)(h)=γ(h)~f(t(h))$ 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 $α\in \widehat{H}$ and an unimodular complex number $a$ such that $(Tf)(h)=a\cdot α(h)~\cdot f(t(h-h_0))\text{ for all }h\in H_0\text{ and for all }f\in C(G)\text{.}$