Banach module valued separating maps and automatic continuity (Q1953956)

For algebras \(A\) and \(B\), a linear map \(T:A\to B\) is separating (or disjointness preserving) if \(x\cdot y=0\) implies \(Tx\cdot Ty=0\). When \(A=B=L^p(\mu)\), such maps were considered by \textit{S. Banach} in his classical treatise [\textit{S. Banach}, Théorie des opérations linéaires. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk (1932; Zbl 0005.20901; JFM 58.0420.01)]; other (more general) cases have been studied extensively in the literature. In this paper, the authors introduce the notion of (hyper) cozero set of elements in a Banach module and extend the notion of separating maps to Banach modules. For a unital commutative Banach algebra \(A\), a unital commutative semisimple regular Banach algebra \(B\) satisfying the Ditkin condition, and a unital, hyper-semisimple, left Banach \(A\)-module \(X\), it is shown that each bijective separating linear map \(T:B\to X\) is continuous and its inverse is also separating.