Strongly zero product determined Banach algebras (Q821023)

The paper under review is concerned with the study of certain classes of Banach algebras from the point of view of being \textit{zero product determined}. More precisely, the following quantitative version of the property ``zero product determined'' is investigated. For the Banach algebra \(A\) there exists a constant \(\alpha\) such 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 \[\sup_{\|a\| =1 = \|b\|} |\varphi(a,b)- \xi(a,b) | \leqslant \alpha \sup_{\|a\| =1 = \|b\|, \; ab=0} | \varphi(a,b)| \] in each of the following cases: \begin{itemize} \item[(i)] \(A\) is a \(C^*\)-algebra; \item[(ii)] \(A = L^1(G)\), where \(G\) is a locally compact group; \item[(iii)] \(A = \mathcal{A}(X)\), the algebra of approximable operators on a Banach space \(X\) which has property \(X\). \end{itemize} In each of the above cases the precise value of \(\alpha\) is given.