Disjointness preserving mappings on BSE Ditkin algebras

Abstract

Let A and B be regular Banach function algebras. A linear map T defined from A into B is said to be disjointness preserving or sepa-We prove that if there exists a disjointness preserving bijection between two BSE Ditkin algebras with a BAI, then they are isomorphic as algebras. As a corollary we can deduce that two of these algebras are algebraically isomorphic if there exists a surjective isometry between them for the supremum norm.

Separating maps between commutative Banach algebras

“…Subsequently, imposing the existence of an approximate identity norm bounded by 1, it is proven that the existence of a bijective separating map also leads to this fact that underlying algebras are algebraically isomorphic. A similar result has been proven in [14] for BSE-algebras satisfying Ditkin's condition.…”