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Letandbe two nests withandcomplemented inandwheneverand, respectively. Assume thatis a unital additive surjection. In this paper, all additive mapspreserving commutativity up to a factor() in both directions (i.e.satisfiesif and only iffor all) are characterized. [ABSTRACT FROM AUTHOR]

Abstract: Let be a nest algebra associated with the nest on a (real or complex) Banach space . Assume that every is complemented whenever . Let be an additive map. It is shown that the following three conditions are equivalent: (1) is derivable at zero point, i.e., whenever ; (2) is Jordan derivable at zero point, i.e., whenever ; (3) has the form for some additive derivation and some scalar . It is also shown that is generalized derivable at zero point, i.e., whenever , if and only if is an additive generalized derivation. Finer characterizations of above maps are given for the case . [Copyright &y& Elsevier]

Abstract: Let and be nest algebras associated with the nests and on Banach Spaces. Assume that and are complemented whenever and . Let be a unital additive surjection. It is shown that preserves Jordan zero-products in both directions, that is , if and only if 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. [Copyright &y& Elsevier]