In this paper we continue our work in [20]. For a Banach algebra A with a character ϕ 2 Δ(A), we discuss the relation of ϕ-bi atness and left ϕ-amenability. We show that if a Segal algebra S(G) (S(G)) is ϕ-bi at, then G is an amenable group. Also we show that ϕ-bi atness of a symmetric Segal algebra S(G) is equivalent with amenability of G. We give the notion of bounded character bi at Banach algebras and study its character spaces. We show that for a non-empty totally ordered set I with a smallest element, upper triangular I I-matrix algebra, say UPI (A) is bi at if and only if A is bi at and I is singleton, provided that Δ(A) is non-empty and A has a right identity. Also we give a class of non bi at Banach algebras.
