Jordan and Jordan Higher All-derivable Points of Some Algebras

TL;DR

This paper characterizes Jordan derivable mappings and all-derivable points in certain algebras, extending results to higher derivations and applying them to nest algebras on Banach spaces.

Contribution

It introduces a characterization of Jordan higher derivable mappings and identifies Jordan all-derivable points in nest algebras, generalizing previous results.

Findings

Jordan derivable mappings characterized via Peirce decomposition

Jordan all-derivable points identified for specific bimodules

Application to nest algebras with complemented elements

Abstract

In this paper, we characterize Jordan derivable mappings in terms of Peirce decomposition and determine Jordan all-derivable points for some general bimodules. Then we generalize the results to the case of Jordan higher derivable mappings. An immediate application of our main results shows that for a nest $\mathcal{N}$ on a Banach $X$ with the associated nest algebra $alg\mathcal{N}$, if there exists a non-trivial element in $\mathcal{N}$ which is complemented in $X$, then every $C\in alg\mathcal{N}$ is a Jordan all-derivable point of $L(alg\mathcal{N}, B(X))$ and a Jordan higher all-derivable point of $L(alg\mathcal{N})$.