Cozero containment preserving maps between certain Banach modules

A linear map T: A→B between spaces of functions A and B is called disjointness preserving if the equation coz(f)∩coz(g)=∅ implies coz(Tf)∩coz(Tg)=∅ and is called cozero containment preserving if the containment coz(f)⊂coz(g) implies coz(Tf)⊂coz(Tg), for all f,g∈A. We first generalize the definition of disjointness preserving maps to Banach module cases and then we show that every bijective disjointness preserving map T: A→X is automatically continuous and has a disjointness preserving inverse, where A is a unital commutative semisimple regular Banach algebra satisfying the Ditkin’s condition and X is a unital hyper semisimple Banach left module over a unital commutative Banach algebra. Finally, we prove that for each bijection cozero containment preserving map T: X→A, under certain conditions, T,T^(-1)are disjointness preserving and T^(-1) is continuous.

Keywords:

Banach modules , cozero containment preserving , disjointness.