Strongly zero product determined Banach algebras

TL;DR

This paper provides a quantitative estimate of the zero product determined property for certain Banach algebras, including $C^*$-algebras and group algebras, by establishing bounds relating bilinear functionals to linear functionals.

Contribution

It introduces a constant $oldsymbol{\alpha ext{,}} ext{ }$ for Banach algebras that quantifies how bilinear functionals approximate linear functionals based on zero product behavior.

Findings

Established explicit bounds for $C^*$-algebras.

Extended results to group algebras.

Provided estimates for approximable operators on Banach spaces.

Abstract

$C^*$-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 \[ \sup_{\Vert a\Vert=\Vert b\Vert=1}\vert\varphi(a,b)-\xi(ab)\vert\le \alpha\sup_{\mathclap{\substack{\Vert a\Vert=\Vert b\Vert=1, \\ ab=0}}}\vert\varphi(a,b)\vert. \]