Vali-e-Asr university of Rafsanjan Wavelet and Linear Algebra 2383-1936 7 2 2020 11 01 The Banach algebras with generalized matrix representation جبرهای باناخ دارای نمایش ماتریسی تعمیم یافته 23 29 46689 10.22072/wala.2020.122402.1273 EN S. Barootkoob Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord, Islamic Republic of Iran. Journal Article 2020 02 29 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 $\Big[\begin{array}{cc}<br />A & \ M \\<br />N & \ B%<br />\end{array}%<br />\Big]$.<br />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 $H^1(A,A)=\{0\}$.<br />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}. https://wala.vru.ac.ir/article_46689_16d40a3be745187190a9683cc8a9e1c0.pdf