Additive maps preserving Jordan zero-products on nest algebras

Abstract

Let Alg N and Alg M be nest algebras associated with the nests N and M on Banach Spaces. Assume that N ∈ N and M ∈ M are complemented whenever N - = N and M - = M . Let Φ : Alg N → Alg M be a unital additive surjection. It is shown that Φ preserves Jordan zero-products in both directions, that is Φ ( A ) Φ ( B ) + Φ ( B ) Φ ( A ) = 0 ⇔ AB + BA = 0 , if and only if Φ is either a ring isomorphism or a ring anti-isomorphism. Particularly, all unital additive surjective maps between Hilbert space nest algebras which preserves Jordan zero-products are characterized completely.