Extension of continuous functions on product spaces, Bohr Compactification and Almost Periodic Functions
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912269896515584 |
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| author | Hernández, Salvador |
| author_facet | Hernández, Salvador |
| contents | The Bohr compactification is a well known construction for (topological) groups and semigroups. Recently, this notion has been investigated for arbitrary structures in \cite{har_kun:bohr_discrete} where the Bohr compactification is defined, using a set-theoretical approach, as the maximal compactification which is compatible with the structure involved. Here, we give a characterization of the continuous functions defined on a product space that can be extended continuously to certain compactifications of the product space. As a consequence, the Bohr compactification of an arbitrary topological structure is obtained as the Gelfand space of the commutative Banach algebra of all almost periodic functions. Previously, almost periodic functions $f$ are defined in terms of translates of $f$ with no reference to any compactification of the underlying structure. An application is given to the representation of isometries defined between spaces of almost periodic functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08265 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extension of continuous functions on product spaces, Bohr Compactification and Almost Periodic Functions Hernández, Salvador Functional Analysis General Topology 46E25, 05C38, 15A15, 05A15, 15A18 05A15, 15A18 The Bohr compactification is a well known construction for (topological) groups and semigroups. Recently, this notion has been investigated for arbitrary structures in \cite{har_kun:bohr_discrete} where the Bohr compactification is defined, using a set-theoretical approach, as the maximal compactification which is compatible with the structure involved. Here, we give a characterization of the continuous functions defined on a product space that can be extended continuously to certain compactifications of the product space. As a consequence, the Bohr compactification of an arbitrary topological structure is obtained as the Gelfand space of the commutative Banach algebra of all almost periodic functions. Previously, almost periodic functions $f$ are defined in terms of translates of $f$ with no reference to any compactification of the underlying structure. An application is given to the representation of isometries defined between spaces of almost periodic functions. |
| title | Extension of continuous functions on product spaces, Bohr Compactification and Almost Periodic Functions |
| topic | Functional Analysis General Topology 46E25, 05C38, 15A15, 05A15, 15A18 05A15, 15A18 |
| url | https://arxiv.org/abs/2503.08265 |