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On the multiplier semigroup of a weighted abelian semigroup. (English) Zbl 1439.43003

Summary: Let \((S, \omega)\) be a weighted abelian semigroup. We show that a \(\omega\)-bounded semigroup multiplier on \(S\) is a multiplication by a bounded function on the space of \(\omega\)-bounded generalized semicharacters on \(S\); and discuss a converse. Given a \(\omega\)-bounded multiplier \(\alpha\) on \(S\), we investigate the induced weighted semigroup \((S_\alpha; \omega_\alpha)\). We show that the \(\omega_\alpha\)-bounded generalized semicharacters on \(S_\alpha\) are scalar multiples of \(\omega\)-bounded generalized semicharacters on \(S\). Moreover, if \((S_0, \omega_0)\) is another weighted semigroup formed with some other operation on set \(S\) such that \(\omega_0\)-bounded generalized semicharacters on \(S_0\) are scalar multiples of \(\omega\)-bounded generalized semicharacters on \(S\), then it is shown that \(S_0 = S_\alpha\) under some natural conditions. A number of examples and counter examples are discussed. The paper strengthens the idea that a weighted semigroup provides a semigroup analogue of a normed algebra for which a Gelfand duality may be searched.

MSC:

43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
46J99 Commutative Banach algebras and commutative topological algebras
20M14 Commutative semigroups
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