Lie ring isomorphisms between nest algebras on Banach spaces

Let N and M be nests on Banach spaces X and Y over the (real or complex) field F and let Alg N and Alg M be the associated nest algebras, respectively. It is shown that a map Phi : Alg N -> Alg M is a Lie ring isomorphism (i.e., Phi is additive, Lie multiplicative and bijective) if and only if Phi has the form Phi(A) = TAT(-1) + h(A)I for all A is an element of Alg N or Phi(A) = -TA*T-1 + h(A)I for all A is an element of Alg N, where h is an additive functional vanishing on all commutators and T is an invertible bounded linear or conjugate linear operator when dim X = infinity; T is a bijective tau-linear transformation for some field automorphism T of F when dim X < infinity. (C) 2014 Elsevier Inc. All rights reserved.