  $C^*$-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.
  We are interested in giving a quantitative estimate of this property by
finding, for each Banach algebra $A$ of the above classes, a constant $\alpha$
with the property that for every continuous bilinear functional $\varphi\colon
A \times A\to\mathbb{C}$ there exists a continuous linear functional $\xi$ on
$A$ such that
  \[
  \sup_{\Vert a\Vert=\Vert b\Vert=1}\vert\varphi(a,b)-\xi(ab)\vert\le
  \alpha\sup_{\mathclap{\substack{\Vert a\Vert=\Vert b\Vert=1, \\
ab=0}}}\vert\varphi(a,b)\vert.
  \]
