Action of Pontryagin dual of semilattices grading algebras

TL;DR

This paper explores the relationship between S-graded algebras and their Pontryagin duals for bounded semilattices, extending classical results from finite abelian groups to more general algebraic structures.

Contribution

It investigates how Pontryagin duality applies to S-graded algebras where S is a bounded semilattice, generalizing known duality results beyond finite abelian groups.

Findings

Established duality relations for S-graded algebras with S as a bounded semilattice

Analyzed the role of the Pontryagin dual S* in the structure of S-graded algebras

Extended classical duality results to infinite and non-abelian contexts

Abstract

Let A be a unitary 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 G* 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 semmilattices. Moreover, if S is such a monoid, we are going to investigate the role of S and its Pontryagin dual S* over the algebra A, in the case A is S-graded.