$C^{⁎}$-algebras, group algebras, and the algebra $\mathcal{A}(X)$ of approximable operators on a Banach space X having the bounded approximation property are known to be zero product determined. In this paper we give a quantitative estimate of this property by showing that, for the Banach algebra A, there exists a constant α with the property that for every continuous bilinear functional $\phi :A\times A\to \mathbb{C}$ there exists a continuous linear functional ξ on A such that$\underset{\Vert a\Vert =\Vert b\Vert =1}{\sup }|\phi (a,b)-\xi (ab)|\le \alpha \underset{\begin{array}{c}\hfill \Vert a\Vert =\Vert b\Vert =1,\hfill \\ \hfill ab=0\hfill \end{array}}{\sup }|\phi (a,b)|$ in each of the following cases: (i) A is a $C^{⁎}$-algebra, in which case $\alpha =8$; (ii) $A=L^1(G)$ for a locally compact group G, in which case $\alpha =60\sqrt{27}\frac{1+\sin \frac{\pi }{10}}{1-2\sin \frac{\pi }{10}}$; (iii) $A=\mathcal{A}(X)$ for a Banach space X having property $(\mathbb{A})$ (which is a rather strong approximation property for X), in which case $\alpha =60\sqrt{27}\frac{1+\sin \frac{\pi }{10}}{1-2\sin \frac{\pi }{10}}{C}^2$, where C is a constant associated with the property $(\mathbb{A})$ that we require for X.

$C^{⁎}$-代数、群代数和代数 $\mathcal{一种}(X)$具有有界逼近性质的 Banach 空间X上的逼近算子的数量已知为零乘积确定。在本文中，我们通过证明对于 Banach 代数A存在一个常数α，其性质为对于每个连续双线性泛函，我们给出了这个性质的定量估计$\phi ：\mathrm{一种}\times \mathrm{一种}\to \mathbb{C}$在A上存在一个连续的线性泛函ξ使得$\underset{\Vert \mathrm{一种}\Vert =\Vert 乙\Vert =1}{\mathrm{超}}|\phi (\mathrm{一种},乙)-\xi (\mathrm{一种}乙)|\le \alpha \underset{\begin{array}{c}\hfill \Vert \mathrm{一种}\Vert =\Vert 乙\Vert =1,\hfill \\ \hfill \mathrm{一种}乙=0\hfill \end{array}}{\mathrm{超}}|\phi (\mathrm{一种},乙)|$在以下每种情况下： (i) A是$C^{⁎}$-代数，在这种情况下 $\alpha =8$; (二)$\mathrm{一种}={升}^1(G)$对于局部紧群G，在这种情况下$\alpha =60\sqrt{27}\frac{1+\mathrm{罪}\frac{\pi }{10}}{1-2\mathrm{罪}\frac{\pi }{10}}$; (三)$\mathrm{一种}=\mathcal{一种}(X)$对于具有性质的 Banach 空间X$(\mathbb{一种})$（这是X 的一个相当强的近似属性），在这种情况下$\alpha =60\sqrt{27}\frac{1+\mathrm{罪}\frac{\pi }{10}}{1-2\mathrm{罪}\frac{\pi }{10}}{C}^2$，其中C是与属性关联的常数$(\mathbb{一种})$我们需要X。