Fredholm Separating operators on Regular Banach Function Algebras

For two algebras A and B, a linear map T : A → B is called separating if x·y = 0 implies Tx·Ty = 0 for all x,y ∈ A. The general form and the automatic continuity of Fredholm separating operators between the Banach algebra of continuous functions defined on a locally compact spaces which vanishing at infinity have been studied. In this paper we give some results about Fredholm separating operator on certain regular Banach function algebras. In particular, we consider Fredholm weighted composition operator T : A → B , defined between certain regular Banach function algebras A and B by T f(y) = ω(y)f(h(y)), for f ∈ A and an appropriate function h :Y→ X and for a non-vanishing function ω :Y → C