Maps preserving zero products

A linear map T from a Banach algebra A into another B preserves zero products if T(a)T(b) = 0 whenever a, b is an element of A are such that ab = 0. This paper is mainly concerned with the question of whether every continuous linear surjective map T: A -> B that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras. Our method involves continuous bilinear maps phi: A x A -> X (for some Banach space X) with the property that phi(a, b) = 0 whenever a, b is an element of A are such that ab = 0. We prove that such a map necessarily satisfies phi(a mu, b) = phi(a, mu b) for all a, b is an element of A and for all mu from the closure with respect to the strong operator topology of the subalgebra of M(A) (the multiplier algebra of A) generated by doubly power-bounded elements of M(A). This method is also shown to be useful for characterizing derivations through the zero products.