Additive Jordan isomorphisms of nest algebras on normed spaces

Let X be a real or complex Banach space. Let algN and algMbe two nest algebras on X. Suppose that φ is an additive bijective mapping from algN onto algMsuch that φ(A2) = φ(A)2 for every A ∈ algN. Then φ is either a ring isomorphism or a ring anti-isomorphism. Moreover, if X is a real space or an infinite dimensional complex space, then there exists a continuous (conjugate) linear bijective mapping T such that either φ(A) = TAT −1 for every A ∈ algN or φ(A) = TA∗T −1 for every A ∈ algN.  2003 Published by Elsevier Inc.