The Shilov boundary and the Gelfand spectrum of algebras of generalized analytic functions

Let S be a discrete abelian cancellation semigroup with identity. We consider an algebra of generalized analytic functions defined on the semigroup $ \hat S $ of semicharacters of S that is wider than the Arens-Singer algebra. We show that the strong boundary and the Shilov boundary of this algebra are unions of maximal subgroups of $ \hat S $ . If S contains no nontrivial simple ideals, then both boundaries coincide with the character group of S. In this case we calculate the Gelfand spectrum of the algebra under consideration.