Additive Jordan isomorphisms of nest algebras on normed spaces

Abstract

Let X be a real or complex Banach space. Let algN and algM be two nest algebras on X. Suppose that φ is an additive bijective mapping from algN onto algM such 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.