Action of Pontryagin Dual of Semilattices on Graded Algebras

Abstract

Let A be a ℂ-algebra and G be a finite abelian group. Then a G-graded algebra is merely a G-algebra and viceversa because of the fact that G and its group of characters are isomorphic. This fact is no longer true if we substitute G with infinite or non-abelian groups. In this paper we try to obtain similar results for a special class of abelian monoids, i.e., the bounded semilattices. Moreover, if S is such a monoid, we are going to investigate the role of S and its Pontryagin dual over the algebra A, in the case A is S-graded.