On the algebraic structures in $\A_\Phi(G)$

TL;DR

This paper investigates the algebraic properties of Fig ext`a-Talamanca-Herz-Orlicz algebras on locally compact groups, establishing conditions under which these algebras form Banach algebras, are Segal algebras, and characterizing their spectra.

Contribution

It proves that ${\\A}_{\Phi}(G)$ is a Banach algebra under convolution if and only if $G$ is compact, and explores its structure as a Segal algebra and its spectral properties.

Findings

${\A}_{\Phi}(G)$ is a Banach algebra iff $G$ is compact.

${\A}_{\Phi}(G)$ is a Segal algebra.

Character space of ${\A}_{\Phi}(G)$} for compact abelian $G$ is identified with $\widehat{G}$.

Abstract

Let $G$ be a locally compact group and $(\Phi, \Psi)$ be a complementary pair of $N$-functions. In this paper, using the powerful tool of porosity, it is proved that when $G$ is an amenable group, then the Fig\`a-Talamanca-Herz-Orlicz algebra ${\A}_{\Phi}(G)$ is a Banach algebra under convolution product if and only if $G$ is compact. Then it is shown that ${\A}_{\Phi}(G)$ is a Segal algebra, and as a consequence, the amenability of ${\A}_{\Phi}(G)$ and the existence of a bounded approximate identity for ${\A}_{\Phi}(G)$ under the convolution product is discussed. Furthermore, it is shown that for a compact abelian group $G$, the character space of ${\A}_{\Phi}(G)$ under convolution product can be identified with $\widehat{G}$, the dual of $G$.