For a Banach algebra A with a character 2 (A), we investigate the relation of -bi atness and left -amenability. For the Segal algebra S(G), related to a locally compact group G, we show that if S(G) is -bi at, then G is an amenable group. For a symmetric Segal algebra S(G), we show that S(G) is ????bi at if and only if G is amenable. We introduce a new notion of bounded character bi at Banach algebras and study its maximal ideal space. We study the homological proprties of a class of matrix algebras. We show that for a nonempty 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 (A) is non-empty and A has a right identity. Also we give a class of non bi at Banach algebras.
