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.
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.
Let T be a continuous linear operator on a Banach algebra A. We address the question of whether the constant sup{ aT (b)c : a, b, c ∈ A, ab = bc = 0, a = b = c = 1} being small implies that the distance from T to the space Der(A) of all continuous derivations on A is small. We show that this is the case for amenable group algebras. As a consequence, we deduce that Der(L 1 (G)) is hyperreflexive for each amenable group in [SIN].
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