Approximate biprojectivity and $φ$-biflatness of certain Banach algebras

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 $G$.