Closed ideals with bounded $\Delta$-weak approximate identities in some certain Banach algebras

TL;DR

This paper characterizes amenability of locally compact groups through the existence of bounded Δ-weak approximate identities in specific closed ideals of certain Banach algebras, linking algebraic properties to group properties.

Contribution

It establishes a new characterization of group amenability via bounded Δ-weak approximate identities in closed ideals of Banach algebras.

Findings

Amenability characterized by bounded Δ-weak approximate identities

Results apply to Figà-Talamanca-Herz algebra, _p(G)

Similar characterizations for (G) and C_0^w(G)

Abstract

It is shown that a locally compact group $G$ is amenable if and only if some certain closed ideals of the Fig\`{a}-Talamanca-Herz algebra $A_{p}(G)$ admit bounded $\Delta$-weak approximate identities. Also, similar results are obtained for the function algebras $\mathfrak{L}A(G)$ and $C_{0}^{w}(G)$.