Characterization of Lie multiplicative isomorphisms between nest algebras

Abstract

Let $\mathcal{N}$ and $\mathcal{M}$ be nests on Banach spaces X and Y over the real or complex field $\mathbb{F}$, respectively, with the property that if $M \in \mathcal{M}$ such that M ₋ = M, then M is complemented in Y. Let $Alg\mathcal{N}$ and $Alg\mathcal{M}$ be the associated nest algebras. Assume that $\Phi :Alg\mathcal{N} \to Alg\mathcal{M}$ is a bijective map. It is proved that, if dimX = ∞ and if there is a nontrivial element in $\mathcal{N}$ which is complemented in X, then Φ is Lie multiplicative (i.e. Φ([A,B]) = [Φ(A), Φ(B)] for all $A,B \in Alg\mathcal{N}$) if and only if Φ has the form Φ(A) = TAT ⁻¹ + τ(A) for all $A \in Alg\mathcal{N}$ or Φ(A) = −TA*T ⁻¹ + τ(A) for all $A \in Alg\mathcal{N}$, where T is an invertible linear or conjugate linear operator and $\tau :Alg\mathcal{N} \to \mathbb{F}I$ is a map with τ([A,B]) = 0 for all $A,B \in Alg\mathcal{N}$. The Lie multiplicative maps are also characterized for the case dimX < ∞.