Ternary derivations of nest algebras

TL;DR

This paper characterizes ternary derivations of nest algebras on Banach spaces, showing they are all inner and linking their structure to zero product conditions and centralizers.

Contribution

It provides a complete description of ternary derivations on nest algebras, establishing their inner nature and characterizing related maps via zero product conditions.

Findings

Every ternary derivation on nest algebras is inner.

Existence and uniqueness of linear maps forming ternary derivations under zero product conditions.

Characterization of centralizers and derivations through zero products and local derivations.

Abstract

Suppose that $X$ is a (real or complex) Banach space, $dimX \geq 2$, and $\mathcal{N}$ is a nest on $X$, with each $N$ in $\mathcal{N}$ is complemented in $X$ whenever $N_{-}=N$. A ternary derivation of $Alg\mathcal{N}$ is a triple of linear maps $(\gamma, \delta, \tau)$ of $Alg\mathcal{N}$ such that $\gamma(AB)=\delta(A)B +A\tau(B)$ for all $A,B \in Alg\mathcal{N}$. We show that for linear maps $\delta$, $\tau$ on $Alg\mathcal{N}$ there exists a unique linear map $\gamma$ from $Alg\mathcal{N}$ into $Alg\mathcal{N}$ defined by $\gamma(A)=RA+AT$ for some $R$, $T$ in $Alg\mathcal{N}$ such that $(\gamma, \delta, \tau)$ is a ternary derivation of $Alg\mathcal{N}$ if and only if $\delta$, $\tau$ satisfy $\delta(A)B+A\tau(B)=0$ for any $A$,$B$ in $Alg\mathcal{N}$ with $AB=0$. We also prove that every ternary derivation on $Alg\mathcal{N}$ is an inner ternary derivation. Our results are applied to characterize the (right or left) centralizers and derivations through zero products, local right (left) centralizers, right (left) ideal preserving maps and local derivations on nest algebras.