Strongly zero product determined Banach algebras

Abstract

$C^{\ast }$-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 $\phi :A\times A\to \mathbb{C}$ there exists a continuous linear functional $\xi$ on $A$ such that \[ \sup_{\Vert a\Vert=\Vert b\Vert=1}\vertφ(a,b)-ξ(ab)\vert\le α\sup_{\mathclap{\substack{\Vert a\Vert=\Vert b\Vert=1, \\ ab=0}}}\vertφ(a,b)\vert. \]