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

### Volume 145 / 2016

#### Abstract

In the first part of the paper, we investigate the approximate biprojectivity 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 every continuous weight $w$, we show that $L^{1}(G,w)$ is approximate biprojective if and only if $G$ is compact, provided that $w(g)\geq 1$ for every $g\in G$.

In the second part, we study $\phi $-biflatness of some Banach algebras, where $\phi $ is a character. We show that if $S(G)$ is $\phi _{0}$-biflat, then $G$ is an amenable group, where $\phi _{0}$ is the augmentation character on $S(G)$. Finally, we show that the $\phi $-biflatness of $L^{1}(G)^{**}$ implies the amenability of $G$.