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On the Jacobs-de Leeuw-Glicksberg decomposition

Micky Barthmann, Sohail Farhangi, Yulia Kuznetsova

TL;DR

This work extends the Jacobs-de Leeuw-Glicksberg decomposition to JdLG-admissible semigroups acting on Banach spaces, establishing a canonical division into a reversible part $E_r$ and an almost weakly stable part $E_{aws}$ via a bi-invariant mean on $W(S)$ and the weak closure $\mathscr{S}$. It shows $E_r$ is weakly equivalent to a unitary representation on a Hilbert space that further decomposes into finite-dimensional components, while $E_{aws}$ is characterized through invariant means and Følner averages, including flight-function/mean criteria. The paper also develops ultrafilter-based descriptions, proving the projection onto $E_r$ can be realized as ultrafilter limits and that essential idempotents in $\beta S$ yield the same projection in many amenable/bi-amenable contexts. Collectively, these results unify JdLG theory with ultrafilter methods and Følner-analytic tools, broadening applicability to amenable and bi-amenable semigroups and clarifying the structure of both reversible and AWS components.

Abstract

For any JdLG-admissible representation $π$ of a semigroup $S$ on a Banach space $E$, we show that the reversible part is weakly equivalent to a unitary representation on a Hilbert space that decomposes into a direct sum of finite dimensional representations, and we give an alternative characterization of the almost weakly stable part in terms of the unique invariant mean on the space of weakly almost periodic functions. In the case that $S$ is a bi-amenable measured semigroup, we characterize the almost weakly stable part using invariant means and averages along Følner sequences. Moreover, we give a description of the unique projection onto the reversible part whose kernel is the almost weakly stable part in terms of ultrafilters.

On the Jacobs-de Leeuw-Glicksberg decomposition

TL;DR

This work extends the Jacobs-de Leeuw-Glicksberg decomposition to JdLG-admissible semigroups acting on Banach spaces, establishing a canonical division into a reversible part and an almost weakly stable part via a bi-invariant mean on and the weak closure . It shows is weakly equivalent to a unitary representation on a Hilbert space that further decomposes into finite-dimensional components, while is characterized through invariant means and Følner averages, including flight-function/mean criteria. The paper also develops ultrafilter-based descriptions, proving the projection onto can be realized as ultrafilter limits and that essential idempotents in yield the same projection in many amenable/bi-amenable contexts. Collectively, these results unify JdLG theory with ultrafilter methods and Følner-analytic tools, broadening applicability to amenable and bi-amenable semigroups and clarifying the structure of both reversible and AWS components.

Abstract

For any JdLG-admissible representation of a semigroup on a Banach space , we show that the reversible part is weakly equivalent to a unitary representation on a Hilbert space that decomposes into a direct sum of finite dimensional representations, and we give an alternative characterization of the almost weakly stable part in terms of the unique invariant mean on the space of weakly almost periodic functions. In the case that is a bi-amenable measured semigroup, we characterize the almost weakly stable part using invariant means and averages along Følner sequences. Moreover, we give a description of the unique projection onto the reversible part whose kernel is the almost weakly stable part in terms of ultrafilters.

Paper Structure

This paper contains 10 sections, 13 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topological semigroups and weak almost periodicity
  4. JdLG-admissible semigroups
  5. Amenable semigroups and Fø lner nets
  6. Representations of semigroups
  7. Ultrafilters and the algebra of the Stone-Čech compactification
  8. The reversible part of the JdLG-decomposition
  9. The almost weakly stable part of the JdLG-decomposition
  10. The JdLG decomposition via ultrafilters

Key Result

Theorem 1.1

Let $S$ be a semitopological semigroup which admits a bi-invariant mean $m$ on $W(S)$, the space of weakly almost periodic functions on $S$. Let $E$ be a Banach space and let $\pi$ be a relatively weakly compact representation of $S$ on $E$. For $\mathscr{S} = c\ell_\sigma(\pi(S))$, the weak closure and both of $E_r$ and $E_{\text{aws}}$ are closed $\mathscr{S}$-invariant subspaces.

Theorems & Definitions (26)

  • Theorem 1.1: cf. AnalysisOnSemigroups
  • Theorem 1.2: cf. OTAoET
  • Remark
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Theorem 3.1: cf. OTAoET, Proposition 16.29.
  • Corollary 3.2
  • proof
  • ...and 16 more