Lie ring isomorphisms between nest algebras on Banach spaces

Abstract

Let $\mathcal{N}$ and $\mathcal{M}$ be nests on Banach spaces $X$ and $Y$ over the (real or complex) field $\mathbb{F}$ and let $\mbox{\rm Alg}{\mathcal N}$ and $\mbox{\rm Alg}{\mathcal M}$ be the associated nest algebras, respectively. It is shown that a map $\Phi :\mathrm{Alg}\mathcal{N}\to \mathrm{Alg}\mathcal{M}$ is a Lie ring isomorphism (i.e., $\Phi$ is additive, Lie multiplicative and bijective) if and only if $\Phi$ has the form $\Phi (A)=TA{T}^{-1}+h(A)I$ for all $A\in \mbox{\rm Alg}{\mathcal N}$ or $\Phi (A)=-T{A}^{\ast }{T}^{-1}+h(A)I$ for all $A\in \mbox{\rm Alg}{\mathcal N}$, where $h$ is an additive functional vanishing on all commutators and $T$ is an invertible bounded linear or conjugate linear operator when $\dim X=\infty$; $T$ is a bijective $\tau$-linear transformation for some field automorphism $\tau$ of $\mathbb{F}$ when $\dim X<\infty$.

27 pages