Articles

Science China Mathematics

, Volume 54, Issue 11, pp 2453-2462

First online:

Characterization of Lie multiplicative isomorphisms between nest algebras

  • XiaoFei QiAffiliated withDepartment of Mathematics, Shanxi University
  • , JinChuan HouAffiliated withDepartment of Mathematics, Shanxi UniversityDepartment of Mathematics, Taiyuan University of Technology Email author 

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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 < ∞.

Keywords

Banach spaces nest algebras Lie ring isomorphisms Lie multiplicative maps

MSC(2000)

47L35 16W10