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

Abstract

In this paper we are going to investigate the approximate biprojectivity and the $\varphi$-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 $w\ge 1$, we show that $L^1(G,w)$ is a approximate biprojective if and only if $G$ is compact. We study $\varphi$-biflatness of some Banach algebras, where $\varphi :A\to \mathbb{C}$ is a multiplicative linear functional. We show that if $S(G)$ is $\varphi$-biflat, then $G$ is amenable group. Also we show that the $\varphi$-biflatness of $L^1(G{)}^{\ast \ast }$ implies the amenability of $G$.