Compactification of bounded semigroup representations

TL;DR

This paper develops a generalized method for compactifying bounded semigroup representations, extending classical theories to broader classes of functions and providing an abstract framework for almost periodicity in semigroup theory.

Contribution

It introduces a novel approach to compactify semigroup representations in $L(X)$, generalizing deLeeuw-Glicksberg theory and refining the setting for bounded $C_0$-semigroups.

Findings

Provides a unified abstract framework for almost periodicity

Extends classical compactification methods to broader semigroup classes

Refines the theory specifically for bounded $C_0$-semigroups

Abstract

The given study uses the methods to identify compactifications of semigroups $S\subset L(X),$ which reside in the space $L(X).$ This method generalizes in some sense the deLeeuw-Glicksberg-Theory to a greater class of functions. The approach provides an abstract approach to several notions of almost periodicity, which mainly involving right semitopological semigroups \cite{RuppertLNM}, and the adjoint theory. Moreover, the given setting is refined to the case of bounded $C_0-$semigroups.