The 𝑙₁-algebra of a commutative semigroup

Abstract

of analytic devices all play a role. It will be seen, however, that the functionals making up the algebras under study have obvious representations as integrals with respect to countably additive, totally finite measures that vanish except on countable sets. Thus the refinements of measure and integration theory are avoided, nothing more recondite than infinite series being needed for all of the integral computations employed. The great majority of our results deal only with the commutative case: in view of the genuine difficulties connected with harmonic analysis even on noncommutative groups and the obvious fact that semigroups are less tractable than groups, no apology is perhaps required for this. The present paper may be described as an introduction to harmonic analysis on discrete commutative semigroups. It will no doubt be noted that we state no theorems concerning Tauberian theorems, analogues of the Pontryagin duality theorem, or the Silov boundary. We hope to deal with these topics in a subsequent communication. 1.2. Throughout this paper, we use the terminology, notions, and results of [3 ]. A few points, however, require re-statement, and we make a few new definitions. The symbol - is used between semigroups, algebras, etc., to denote the existence of a 1-to-1 correspondence 7r preserving all operations of both systems; and if a product, say, is defined only for some pairs, then the equality 7r(xy) =7r(x)7r(y) is to hold whenever either side is defined. A homomorphism of a semigroup G onto a semigroup H is a single-valued mapping ,u of G onto H such that ,u(xy) =,(x),u(y) for all x, yCG. 1.3. A multiplicative function on a semigroup G is any complex function f on G satisfying the functional equation f(xy) =f(x)f(y) for all x, yCG. A