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.