A note on left $\varphi$-biflat Banach algebras

Abstract

In this paper, we study the notion of $\varphi$-biflatness for some Banach algebras, where $\varphi$ is a non-zero multiplicative linear functional. We show that the Segal algebra $S(G)$ is left $\varphi$-biflat if and only if $G$ is amenable. Also, we characterize left $\varphi$-biflatness of semigroup algebra ${\ell }^1(S)$ in the term of biflatness, where $S$ is a Clifford semigroup.