The Banach algebras with generalized matrix representation

Abstract

A Banach algebra $mathfrak{A}$ has a generalized Matrix representation if there exist the algebras $A, B$, $(A,B)$-module $M$ and $(B,A)$-module $N$ such that $mathfrak{A}$ is isomorphic to the generalized matrix Banach algebra $Big[begin{array}{cc}A & M \N & B%end{array}%Big]$.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 cite{D}.