Weakly almost periodic topologies, idempotents and ideals

TL;DR

This paper explores the connections between weakly almost periodic topologies, central idempotents, and ideals in the algebra of weakly almost periodic functions, providing new decompositions of representations and extending to Fourier-Stieltjes algebras.

Contribution

It introduces new relationships and decompositions involving weakly almost periodic topologies, idempotents, and ideals, extending existing theories to broader classes of topologies and algebras.

Findings

Decompositions of weakly almost periodic representations.

Relationships between topologies, idempotents, and ideals.

Extensions to Fourier-Stieltjes algebras.

Abstract

Let (G,tau_G) be a topological group. We establish relationships between weakly almost periodic topologies on G coarser than tau_G, central idempotents in the weakly almost periodic compactification G^W, and certain ideals in the algebra of weakly almost periodic functions W(G). We gain decompositions of weakly almost periodic representations, generalizing many from the literature. We look at the role of pre-locally compact topologies, unitarizable topologies, and extend or decompositions to Fourier-Stieltjes algebras B(G).