Separating maps between commutative Banach algebras

TL;DR

This paper investigates the properties of separating maps between commutative Banach algebras, establishing conditions under which such maps are continuous and induce homeomorphisms of maximal ideal spaces, with implications for algebraic isomorphisms.

Contribution

It introduces a new class of regular Tauberian algebras and characterizes bijective separating maps, showing their continuity and topological implications.

Findings

Bijective separating maps are continuous.

Maximal ideal spaces are homeomorphic under these maps.

Additional conditions yield algebraic isomorphisms.

Abstract

Let $\cal A$ and $\cal B$ be Banach algebras. A linear map $T:{\cal A} \rightarrow {\cal B}$ is called separating or disjointness preserving if $ab=0$ implies $Ta\;Tb = 0$ for all $a,b\in {\cal A}$. In this paper, we study a new class of regular Tauberian algebras and prove that some well-known Banach algebras in harmonic analysis belong to this class. We show that a bijective separating map between these algebras turns out to be continuous and the maximal ideal spaces of underlying algebras are homeomorphic. By imposing extra conditions on these algebras, we find a more thorough characterization of separating maps. The existence of a bijective separating map also leads to the existence of an algebraic isomorphism in some cases.