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. Let AlgN and AlgM be the associated nest algebras, respectively. It is shown that a map Φ:AlgNâ†'Alg.M is a Lie ring isomorphism (i.e.,Φis additive, Lie multiplicative and bijective) if and only if Φ(A)=TAT-1+h(A)l for all A∈e AlgN or Φ(A)=-TA*T-1+h(A)I for all A∈AlgAN, where h is an additive functional vanishing on all commutators and T is a bijective Ï„-linear transformation for some field automorphism Ï„ of F when dim X<∞; T is an invertible bounded linear or conjugate linear operator when dim X=∞; if F=R, T is linear.