Group homomorphisms induced by isometries
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910841218007040 |
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| author | Hernández, Salvador |
| author_facet | Hernández, Salvador |
| contents | 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 $(Tf)(h)=a\cdot α(h)~\cdot f(t(h-h_0))\text{ for
all }h\in H_0\text{ and for all }f\in C(G)\text{.}$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16712 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Group homomorphisms induced by isometries Hernández, Salvador Functional Analysis General Topology 22D35, 43A60, 46E15, 54C40, 43A40 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 $(Tf)(h)=a\cdot α(h)~\cdot f(t(h-h_0))\text{ for all }h\in H_0\text{ and for all }f\in C(G)\text{.}$ |
| title | Group homomorphisms induced by isometries |
| topic | Functional Analysis General Topology 22D35, 43A60, 46E15, 54C40, 43A40 |
| url | https://arxiv.org/abs/2502.16712 |