BSE-property for some certain Segal and Banach algebras

TL;DR

This paper characterizes when certain Segal and Banach algebras possess the BSE-property, linking it to bounded weak approximate identities and group properties like weak amenability.

Contribution

It provides necessary and sufficient conditions for BSE-property in abstract Segal algebras within Banach algebras and Fourier algebras, including new classes related to local functions.

Findings

Abstract Segal algebras in Banach algebras are BSE if they have bounded weak approximate identities.

A class of Segal algebras in Fourier algebras are BSE if and only if they have bounded weak approximate identities.

For discrete groups, $A_{cb}(G)$ is BSE if and only if G is weakly amenable.

Abstract

For a commutative semi-simple Banach algebra ${A}$ which is an ideal in its second dual we give a necessary and sufficient condition for an essential abstract Segal algebra in ${A}$ to be a BSE-algebra. We show that a large class of abstract Segal algebras in the Fourier algebra $A(G)$ of a locally compact group $G$ are BSE-algebra if and only if they have bounded weak approximate identities. Also, in the case that $G$ is discrete we show that $A_{\rm cb}(G)$ is a BSE-algebra if and only if $G$ is weakly amenable. We study the BSE-property of some certain Segal algebras implemented by local functions that were recently introduced by J. Inoue and S.-E. Takahasi. Finally we give a similar construction for the group algebra implemented by a measurable and sub-multiplicative function.