Automatic Continuity of Higher Derivations

Madjid Mirzavaziri ; Elahe Omidvar Tehrani

Bulletin of the Malaysian Mathematical Society, Tome 35 (2012) no. 4

Résumé

Let $\mathcal A$ and $\mathcal B$ be two algebras. A sequence $\{d_n\}$ of linear mappings from $\mathcal A$ into $\mathcal B$ is called a higher derivation if $d_n(a_1a_2)=\sum_{k=0}^n d_k(a_1)d_{n-k}(a_2)$ for each $a_1,a_2\in{\mathcal A}$ and each nonnegative integer $n$. In this paper, we show that if $\{d_n\}$ is a higher derivation from $\mathcal A$ into $\mathcal B$ such that $d_0$ is onto and $\ker(d_0)\subseteq\ker(d_n)~(n\in\mathbb{N})$, then there is a sequence $\{\delta_n\}$ of derivations on ${\mathcal B}$ such that $d_n=\sum_{i=1}^n\left(\sum_{\sum_{j=1}^ir_j=n}\left(\prod_{j=1}^i\frac1{r_j+\ldots+r_i}\right) \delta_{r_1}\ldots\delta_{r_i}d_0\right).$ As a corollary we prove that a higher derivation $\{d_n\}$ from a Banach algebra into a semisimple Banach algebra is continuous provided that $d_0$ is onto and $\ker(d_0)\subseteq\ker(d_n)~(n\in\mathbb{N})$. We also deduce that if $\mathcal A$ is a semisimple Jordan Banach algebra and $\{d_n\}$ is a higher derivation on $\mathcal A$ with $d_0(\mathcal A)=\mathcal A$ and $\ker(d_0)\subseteq\ker(d_n)~(n\in\mathbb{N})$ then $\{d_n\}$ is continuous.

Détail

Classification : 46H40, 16W25, 47B47, 46L57.