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Amenability and Invariant subspaces of the algebra of pseudomeasures

Arvish Dabra, N. Shravan Kumar

TL;DR

The paper develops a structural link between amenability and invariant subspaces of the $Ψ$-pseudomeasure algebra $PM_Ψ(G)$ alongside the Orlicz Figà-Talamanca Herz algebra $A_Φ(G)$. It shows that, for amenable $G$ and a MA-satisfying Young pair $(Φ,Ψ)$, there are precise correspondences between norm-closed topologically invariant subalgebras of $PM_Ψ(G)$ and closed subgroups of $G$, as well as between certain invariant subalgebras of $A_Φ(G)$ and compact subgroups. The methodology combines duality between $A_Φ(G)$ and $PM_Ψ(G)$, properties of topologically introverted subspaces, spectral synthesis, and fixed-point arguments to establish bijections and amenability criteria. These results extend classical Lau–Ülger-type correspondences from $PM_p(G)$ to the Orlicz setting, yielding insights into the harmonic analysis on locally compact groups with general Orlicz structures.

Abstract

Let $G$ be a locally compact group and $(Φ,Ψ)$ a complimentary pair of Young functions. In this article, we consider the Banach algebra of $Ψ$-pseudomeasures $PM_Ψ(G)$ and the Orlicz Figà-Talamanca Herz algebra $A_Φ(G).$ We prove sufficient conditions for a group $G$ to be amenable in terms of the norm closed topologically invariant subspaces of $PM_Ψ(G).$ Further, for an amenable group $G$ with the Young function $Φ$ satisfying the MA condition, we establish a one-to-one correspondence between certain topologically invariant subalgebras of $PM_Ψ(G)$ and the class of closed subgroups of $G.$ Moreover, we prove a similar result for the predual $A_Φ(G)$ and derive a bijection between certain topologically invariant subalgebras of $A_Φ(G)$ and the set of compact subgroups of $G.$

Amenability and Invariant subspaces of the algebra of pseudomeasures

TL;DR

The paper develops a structural link between amenability and invariant subspaces of the -pseudomeasure algebra alongside the Orlicz Figà-Talamanca Herz algebra . It shows that, for amenable and a MA-satisfying Young pair , there are precise correspondences between norm-closed topologically invariant subalgebras of and closed subgroups of , as well as between certain invariant subalgebras of and compact subgroups. The methodology combines duality between and , properties of topologically introverted subspaces, spectral synthesis, and fixed-point arguments to establish bijections and amenability criteria. These results extend classical Lau–Ülger-type correspondences from to the Orlicz setting, yielding insights into the harmonic analysis on locally compact groups with general Orlicz structures.

Abstract

Let be a locally compact group and a complimentary pair of Young functions. In this article, we consider the Banach algebra of -pseudomeasures and the Orlicz Figà-Talamanca Herz algebra We prove sufficient conditions for a group to be amenable in terms of the norm closed topologically invariant subspaces of Further, for an amenable group with the Young function satisfying the MA condition, we establish a one-to-one correspondence between certain topologically invariant subalgebras of and the class of closed subgroups of Moreover, we prove a similar result for the predual and derive a bijection between certain topologically invariant subalgebras of and the set of compact subgroups of
Paper Structure (5 sections, 13 theorems, 58 equations)

This paper contains 5 sections, 13 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Amenability and Invariant subspaces of $PM_\Psi(G)$
  4. Invariant subalgebras of $PM_\Psi(G)$ and closed subgroups of $G$
  5. Invariant subalgebras of $A_\Phi(G)$ and compact subgroups of $G$

Key Result

Lemma 3.1

For a norm closed topologically invariant subspace $X$ of $PM_\Psi(G),$ consider the map $\eta: (\overline{A_\Phi\cdot X})^\ast \to \mathcal{M}(X,PM_\Psi)$ defined by $\eta(m) := m_L,$ where $m_L(\varphi) = m \odot \varphi$ for $\varphi \in X$ and $m \in (\overline{A_\Phi \cdot X})^\ast.$ Then $\eta for all $u \in A_\Phi(G).$ Further, if $X$ is also topologically introverted, then $\eta$ is into $

Theorems & Definitions (25)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Corollary 3.4
  • proof
  • Theorem 3.5
  • proof
  • Lemma 4.1
  • ...and 15 more