A linear map \(\Phi:A\to B\) between Banach function algebras \(A\) and \(B\) is said to be disjointness-preserving if \(\Phi(a)\Phi(b)=0\) for all \(a\), \(b\in A\) such that \(ab=0\). The paper is devoted to a description of such maps on a variety of significant Banach function algebras associated with a locally compact group \(G\) such as the Figà-Talamanca-Herz algebra \(A_p(G)\) and the Figá-Talamanca-Herz-Lebesgue algebra \(A_p^q(G)\) for \(p\in ]1,\infty[\) and \(q\in [1,\infty[\).
For this sake the authors introduce a certain property of commutative Banach algebras which they call property O\({\mathbb B}\), and show that a variety of important Banach algebras in harmonic analysis have this property, namely \(A_p(G)\) together with its quotient \(A_p(E)\) for any locally compact group \(G\) and \(E\subset G\) closed, and \(A_p^q(G)\) together with its quotient \(A_p^q(E)\) whenever the group \(G\) is such that \(A_p(G)\) has a certain approximate identity (which holds for \(G\) amenable) and \(E\subset G\) closed.
The main result of the paper asserts that every bounded disjointness-preserving linear map \(\Phi:A\to B\) from a commutative Banach algebra \(A\) with the property O\({\mathbb B}\) into any semisimple, commutative Banach algebra \(B\) is a weighted composition map.