A Note on Extending Semicharacters on Semigroups

The purpose of this note is to give a necessary and sufficient condition for a semicharacter on a subsemigroup of a commutative semigroup G to be extendable to a semicharacter on G. A bounded complex function x on a semigroup G is called a semicharacter of G if x(x)9^0 for some xEG and x(xy) = x(%)x(y) ior all x, yEG. Semicharacters were introduced by Hewitt and Zuckerman [l] and in a slightly different form by §. Schwarz, [4] and [S]. See also [2]. If X is a semicharacter on G, then | x(x) | fk 1 ior all x£G. We will write a I b if there exists an xEG such that ax = b. We note that our theorem generalizes [6, Lemma 1, p. 94], and is related to [3, Lemma 1, p. 364]. Some remarks on the impossibility of generalizing our theorem appear at the end of the paper.