Lie ring isomorphisms between nest algebras on Banach spaces *

摘要

Let N and M be nests on Banach spaces X and Y over the (real or complex) field F and let AlgN and Alg M be the associated nest algebras, respectively. It is shown that a map Φ: AlgN → Alg M is a Lie ring isomorphism (i.e., Φ is additive, Lie multiplicative and bijective) if and only if Φ has the form Φ(A) = TAT~(-1) + h(A)I for all A ∈ AlgN or Φ(A) = ?TA~*T~(-1) + h(A)I for all A ∈ AlgN, where h is an additive functional vanishing on all commutators and T is an invertible bounded linear or conjugate linear operator when dim X = ∞; T is a bijective τ-linear transformation for some field automorphism τ of F when dim X < ∞.