Extension of characters on *-semigroups
A character on an abelian semigroup S with an involution \((x\to x^ *)\) is, by definition, a nonzero homomorphism \(\rho\) : \(S\to {\mathbb{C}}\) satisfying \(\rho (x^ *)=\overline{\rho (x)}\). Necessary and sufficient conditions are given for a fixed character, or every character, on a given *-subsemigroup T of S to be extendible to a character on S. These conditions are similar to those discovered by \textit{K. A. Ross} [Proc. Am. Math. Soc. 10, 579-583 (1959; Zbl 0090.017)] and by \textit{P. Hill} [J. Algebra 5, 16-24 (1967; Zbl 0146.031)] in the case of complex homomorphisms on abelian semigroups without involution. It is shown that any countable (resp. infinite) *-subsemigroup \(T_ 0\) of S is contained in a *-subsemigroup T of S which is countable (resp. equipotent to \(T_ 0)\) and such that every character on T extends to a character on S.
scientific article; zbMATH DE number 4023583