Multiplicative *-Lie type higher derivations of standard operator algebras

Abstract

Let $\mathfrak{A}$ be a standard operator algebra on an infinite dimensional complex Hilbert space $\mathcal{H}$ containing identity operator I. Let $p_n(X_1,X_2,\cdots ,X_n)$ be the polynomial defined by n indeterminates $X_1,\cdots ,X_n$ and their multiple *-Lie products and $\mathbb{N}$ be the set of non-negative integers. In this paper, it is shown that if $\mathfrak{A}$ is closed under the adjoint operation and $\mathcal{D}={\left\{d_m\right\}}_{m\in \mathbb{N}}$ is the family of mappings $d_m:\mathfrak{A}\to \mathcal{B}(\mathcal{H})$ such that $d_0=i{d}_{\mathfrak{A}},$ the identity map on $\mathfrak{A}$ satisfying $$d_m(p_n({\mathcal{U}}_1,{\mathcal{U}}_2,\cdots ,{\mathcal{U}}_n))=\sum _{i_1+i_2+\cdots +i_n=m}{p}_n(d_{i_1}({\mathcal{U}}_1),d_{i_2}({\mathcal{U}}_2),\cdots ,d_{i_n}({\mathcal{U}}_n))$$ for all ${\mathcal{U}}_1,{\mathcal{U}}_2,\cdots ,{\mathcal{U}}_n\in \mathfrak{A}$ and for each $m\in \mathbb{N}$, then $\mathcal{D}={\left\{d_m\right\}}_{m\in \mathbb{N}}$ is an additive *-higher derivation. Moreover, $\mathcal{D}$ is inner.

Keywords:

• Multiplicative *-Lie-type derivation
• multiplicative *-Lie-type higher derivation
• standard operator algebra

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 47B47
• 46L10