On zero product determined Banach algebras

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>۰ 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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جبر باناخ ، معین ضرب صفر ، خودتوان ، همانی تقریبی