Characterizations of Lie Higher Derivations on J-Subspace Lattice Algebras

TL;DR

This paper characterizes the structure of Lie higher derivations on J-subspace lattice algebras, extending understanding of their behavior under specific algebraic conditions.

Contribution

It provides a detailed characterization of Lie higher derivations on J-subspace lattice algebras, including cases involving generalized Lie brackets with scalar parameters.

Findings

Characterization of Lie higher derivations satisfying specific commutator conditions

Extension to generalized Lie brackets with scalar parameters

Structural description of derivations on J-subspace lattice algebras

Abstract

Let $\mathcal{L}$ be a $\mathcal{J}$-subspace lattice on a Banach space $X$ over the real or complex field $\mathbb{F}$ and $ \mathrm{Alg}\mathcal{L}$ be the associated $\mathcal{J}$-subspace lattice algebras. In this paper, we characterize the structure of a family $\{L_n\}_{n=0}^{\infty}: \mathrm{Alg}\mathcal{L}\rightarrow \mathrm{Alg}\mathcal{L}$ of linear mappings satisfying the condition $$L_n([A, B])=\sum_{i+j=n}[L_i(A), L_j(B)]$$ for any $A, B\in\mathrm{Alg}\mathcal{L}$ with $AB = 0$. Moreover, the family $\{L_n\}_{n=0}^{\infty}: \mathrm{Alg}\mathcal{L}\rightarrow \mathrm{Alg}\mathcal{L}$ of linear mappings satisfying $L_n([A, B]_{\xi})=\sum_{i+j=n}[L_i(A), L_j(B)]_{\xi}$ for any $A, B\in\mathrm{Alg}\mathcal{L}$ with $AB = 0$ and $1\neq \xi\in \mathbb{F}$ is also considered in the current work.