On zero product determined Banach algebras

Document Type : Research Paper


Department of Mathematics, Ayatollah Borujerdi University, Borujerd, Iran



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.


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