Multiplicative Lie triple derivations on standard operator algebras

@article{Wani2021,
abstract = {Let $\mathcal \{X\}$ be a Banach space of dimension $n>1$ and $\mathfrak \{A\} \subset \mathcal \{B\}(\mathcal \{X\})$ be a standard operator algebra. In the present paper it is shown that if a mapping $d:\mathfrak \{A\} \rightarrow \mathfrak \{A\}$ (not necessarily linear) satisfies \[d([[U,V],W])=[[d(U),V],W]+[[U,d(V)],W]+[[U,V],d(W)]\] for all $U, V, W \in \mathfrak \{A\}$, then $d=\psi +\tau $, where $\psi $ is an additive derivation of $\mathfrak \{A\}$ and $\tau : \mathfrak \{A\} \rightarrow \mathbb \{F\}I$ vanishes at second commutator $[[U,V],W]$ for all $U, V, W \in \mathfrak \{A\}$. Moreover, if $d$ is linear and satisfies the above relation, then there exists an operator $S\in \mathfrak \{A\}$ and a linear mapping $\tau $ from $\mathfrak \{A\}$ into $\mathbb \{F\}I$ satisfying $\tau ([[U,V],W])=0$ for all $U, V, W \in \mathfrak \{A\}$, such that $d(U)=SU-US+\tau (U)$ for all $U\in \mathfrak \{A\}$.},
author = {Wani, Bilal Ahmad},
journal = {Communications in Mathematics},
keywords = {Multiplicative Lie derivation; multiplicative Lie triple derivation; standard operator algebra},
language = {eng},
number = {3},
pages = {357-369},
publisher = {University of Ostrava},
title = {Multiplicative Lie triple derivations on standard operator algebras},
url = {http://eudml.org/doc/298199},
volume = {29},
year = {2021},
}

TY - JOUR
AU - Wani, Bilal Ahmad
TI - Multiplicative Lie triple derivations on standard operator algebras
JO - Communications in Mathematics
PY - 2021
PB - University of Ostrava
VL - 29
IS - 3
SP - 357
EP - 369
AB - Let $\mathcal {X}$ be a Banach space of dimension $n>1$ and $\mathfrak {A} \subset \mathcal {B}(\mathcal {X})$ be a standard operator algebra. In the present paper it is shown that if a mapping $d:\mathfrak {A} \rightarrow \mathfrak {A}$ (not necessarily linear) satisfies \[d([[U,V],W])=[[d(U),V],W]+[[U,d(V)],W]+[[U,V],d(W)]\] for all $U, V, W \in \mathfrak {A}$, then $d=\psi +\tau $, where $\psi $ is an additive derivation of $\mathfrak {A}$ and $\tau : \mathfrak {A} \rightarrow \mathbb {F}I$ vanishes at second commutator $[[U,V],W]$ for all $U, V, W \in \mathfrak {A}$. Moreover, if $d$ is linear and satisfies the above relation, then there exists an operator $S\in \mathfrak {A}$ and a linear mapping $\tau $ from $\mathfrak {A}$ into $\mathbb {F}I$ satisfying $\tau ([[U,V],W])=0$ for all $U, V, W \in \mathfrak {A}$, such that $d(U)=SU-US+\tau (U)$ for all $U\in \mathfrak {A}$.
LA - eng
KW - Multiplicative Lie derivation; multiplicative Lie triple derivation; standard operator algebra
UR - http://eudml.org/doc/298199
ER -