Separating maps between commutative Banach algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be Banach algebras. A linear map $T:{\cal A} \rightarrow {\cal B}$iscalledseparatingordisjointnesspreservingif$ab=0$implies$Ta;Tb = 0$forall$a,b\in {\cal A}$. 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.