Abstract. A linear map T from a Banach algebra A into another B preserves zero products if T (a)T (b) = 0 whenever a, b ∈ 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 φ : A × A → X (for some Banach space X) with the property that φ(a, b) = 0 whenever a, b ∈ A are such that ab = 0. We prove that such a map necessarily satisfies φ(aµ, b) = φ(a, µb) for all a, b ∈ A and for all µ 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.Introduction. S. Banach [4] was the first to describe isometries on L p ([0, 1]) with p = 2. Although Banach did not give the full proof for this case (this was provided by J. Lamperti [26]), he made the key observation that isometries on L p ([0, 1]) must take functions with disjoint support into functions with disjoint support. This property arises in a variety of situations and has been considered by several authors. For example, in the theory of Banach lattices there is an extensive literature about linear maps T : X → Y , where X and Y are Banach lattices, with the property that |T (x)| ∧ |T (y)| = 0 whenever x, y ∈ X are such that |x| ∧ |y| = 0. Such maps are usually called disjointness preserving operators or d-homomorphisms. The reader interested in this setting is referred to the monograph [1]. The concept of a disjointness preserving operator was exported to function algebras by E. Beckenstein and L. Narici (see [5] for general information).

The main theorem states that a bounded linear operator h from a unital C * -algebra A into a unital Banach algebra B must be a homomorphism provided that h(1) = 1 and the following condition holds: if x, y, z ∈ A are such that xy = yz = 0, then h(x)h(y)h(z) = 0. This theorem covers various known results; in particular it yields Johnson's theorem on local derivations.

Let A and B be C * -algebras, let X be an essential Banach A-bimodule and let T : A → B and S : A → X be continuous linear maps with T surjective. Suppose that T (a)T (b) + T (b)T (a) = 0 and S(a)b + bS(a) + aS(b) + S(b)a = 0 whenever a, b ∈ A are such that ab = ba = 0. We prove that then T = wΦ and S = D + Ψ , where w lies in the centre of the multiplier algebra of B, Φ : A → B is a Jordan epimorphism, D : A → X is a derivation and Ψ : A → X is a bimodule homomorphism.

The main theorem states that all Lie and Jordan derivations from a von Neumann algebra $\frak{A}$ into any Banach $\frak{A}$-bimodule are standard. Moreover, this statement is proved for some other classes of algebras. Our approach is based on the algebraic theory of functional identities and the strong degree, and is combined with analytic tools.

A continuous linear map T from a Banach algebra A into another B approximately preserves the zero products if T (a)T (b) ≤ α a b (a, b ∈ A, ab = 0) for some small positive α. This paper is mainly concerned with the question of whether any continuous linear surjective map T : A → B that approximately preserves the zero products is close to a continuous homomorphism from A onto B with respect to the operator norm. We show that this is indeed the case for amenable group algebras.