The Banach algebras with generalized matrix representation

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^۱(A,A)=\{۰\}.This implies that there is a unital Banach algebra A without any triangular matrix representation such that H^۱(A,A)=\{۰\}  and gives a negative answer to the open question of \cite{D}.

Keywords:

جبرباناخ , عنصر خودتوان , جبر باناخ ماتریسی تعمیم یافته