Wavelet and Linear Algebra

Abstract Let $\mathcal{A}$ be a Banach algebra with a left approximate identity.
    In this paper, under each of the following conditions, we prove that $\mathcal{A}$ is zero product determined.
    
    (i) For every continuous bilinear mapping $\phi$ from ${\mathcal A}\times {\mathcal A}$ into ${\mathcal X}$, where ${\mathcal X}$ is a Banach space, there exists $k>0$ such that 
    $\Vert \phi(a,b)\Vert\leq k \Vert ab\Vert$, for all $a,b\in\mathcal{A}$.
    
    (ii) $\mathcal{A}$ is generated by idempotents.

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}.