Biflatness of Abstract Segal Algebras Based on Characters

In this paper, we investigate and study the notion of left -biflatness of abstract Segal algebras, where  is a character on Banach algebra. Precisely, we give a necessary and sufficient condition for left -biflatness of abstract Segal algebras equipped with a left approximate identity. As an application, we show that if  is a Segal algebra on the locally group  and  is a character, then  is left -biflat if and only if  is amenable. Indeed, this is a generalization of [4, Theorem 3.4]. Moreover, we study the relationship between left -biflatness and inner -amenability and show that if the Banach algebra  is inner -amenable, then the notions of left -biflatness and left -amenability are equivalent.