Approximate biprojectivity and $\phi$-biflatness of certain Banach algebras

TL;DR

This paper investigates the approximate biprojectivity and $\phi$-biflatness of Banach algebras related to locally compact groups, establishing conditions under which these properties imply group compactness or amenability.

Contribution

It provides new characterizations of approximate biprojectivity and $\phi$-biflatness for Banach algebras associated with locally compact groups, linking these properties to group compactness and amenability.

Findings

Segal algebra $S(G)$ is approximate biprojective iff $G$ is compact

For weighted $L^1$ spaces, approximate biprojectivity iff $G$ is compact

If $S(G)$ is $\phi$-biflat, then $G$ is amenable

Abstract

In this paper we are going to investigate the approximate biprojectivity and the $\phi$-biflatness of some Banach algebras related to the locally compact groups. We show that a Segal algebra $S(G)$ is approximate biprojective if and only if $G$ is compact. Also for a continuous weight $w\geq 1$, we show that $L^{1}(G,w)$ is a approximate biprojective if and only if $G$ is compact. We study $\phi$-biflatness of some Banach algebras, where $\phi:A\rightarrow \mathbb{C}$ is a multiplicative linear functional. We show that if $S(G)$ is $\phi$-biflat, then $G$ is amenable group. Also we show that the $\phi$-biflatness of $L^{1}(G)^{**}$ implies the amenability of $G$.