Multiplicative Lie triple derivations on standard operator algebras

Wani, Bilal Ahmad

Communications in Mathematics, Tome 29 (2021) no. 3, pp. 357-369.

Résumé

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 $U\in \mathfrak {A}$.

Détail

Classification : 16W25, 47B47, 47B48
Keywords: Multiplicative Lie derivation; multiplicative Lie triple derivation; standard operator algebra.