The Banach algebras with generalized matrix representation

A Banach algebra A has a generalized Matrix representation if there exist the algebras A, B, (A, B)-module M and (B, A)- module N such that A is isomorphic to the generalized matrix Banach algebra h A M N B i . In this paper, the algebras with generalized matrix representation will be characterized. Then we show that there is a unital permanently weakly amenable Banach algebra A without generalized matrix representation such that H 1 (A, A) = {0}. This implies that there is a unital Banach algebra A without any triangular matrix representation such that H 1 (A, A) = {0} and gives a negative answer to the open question of [2].