On the Multiplier Semigroup of a Weighted Abelian Semigroup

Abstract

Let (S, ω) be a weighted abelian semigroup. We show that a ω-bounded semigroup multiplier on S is a multiplication by a bounded function on the space of ω-bounded generalized semicharacters on S; and discuss a converse. Given a ω-bounded multiplier α on S, we investigate the induced weighted semigroup (Sα; ωα). We show that the ωα-bounded generalized semicharacters on Sα are scalar multiples of ω-bounded generalized semicharacters on S. Moreover, if (S0, ω0) is another weighted semigroup formed with some other operation on set S such that ω0-bounded generalized semicharacters on S0 are scalar multiples of ω-bounded generalized semicharacters on S, then it is shown that S0 = Sα 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.