Lie ring isomorphisms between nest algebras on Banach spaces

TL;DR

This paper characterizes Lie ring isomorphisms between nest algebras on Banach spaces, showing they are essentially implemented by invertible operators or conjugate linear transformations, with specific forms depending on the space dimension.

Contribution

It provides a complete description of Lie ring isomorphisms between nest algebras on Banach spaces, including finite and infinite-dimensional cases, extending previous understanding of algebraic structure.

Findings

Lie ring isomorphisms have explicit forms involving invertible operators or conjugate linear transformations.

Characterization depends on the dimension of the underlying Banach space.

The results unify finite and infinite-dimensional cases under a common framework.

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 $\Phi:{\rm Alg}{\mathcal N}\rightarrow{\rm Alg}{\mathcal M}$ is a Lie ring isomorphism (i.e., $\Phi$ is additive, Lie multiplicative and bijective) if and only if $\Phi$ has the form $\Phi(A) = TAT^{-1} + h(A)I$ for all $A\in \mbox{\rm Alg}{\mathcal N}$ or $\Phi(A)=-TA^*T^{-1}+h(A)I$ 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 $\dim X<\infty$.