The Banach algebras with generalized matrix representation

‎An 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}‎.

* The formulas are not displayed correctly.