Topology Proceedings 28 No. 2 (2004), pp. 527-540

Extension of continuous functions on product spaces, Bohr compactification and almost periodic functions

Salvador Hernández

The Bohr compactification is a well known construction for (topological) groups and semigroups. Recently, this notion has been investigated for arbitrary structures in [K. Kunen and J. Hart, Fund. Math. (1999)] 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.

Keywords: Bohr compactification, almost periodic functions, product spaces, extension of continuous functions

Mathematics Subject Classification: 05C38 15A15 (05A15 15A18)

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Topology Proceedings, Volume 28, No. 2 (2004)
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