Inner higher derivations on Hilbert C$ ^* $-modules

Résumé

‎Let $ \mathcal{M} $ be a Hilbert C$ ^* $-module‎. ‎We define a product $ \pi_\textbf{e} $ on $ \mathcal{M} $‎, ‎making $ \mathcal{M} $ into a Banach algebra‎.

‎Then we study inner derivations and inner higher derivations of Banach algebra $ (\mathcal{M},\pi_\textbf{e}) $‎. ‎We‎

‎find the general form of the family of inner derivations corresponding to an inner higher derivation on‎

‎unital Banach algebra $ (\mathcal{M},\pi_\textbf{e}) $‎, ‎with the identity element $ 1_\mathcal{M} $‎. ‎We show that if $ \{v_n\}_{n=0}^\infty $ and $ \{w_n\}_{n=0}^\infty $ are sequences in $\mathcal{M}$ such that $ v_0=w_0=1_\mathcal{M} $ and‎

‎$ (\textbf{v}*\textbf{w})_n=(\textbf{w}*\textbf{v})_n=0 $ for all $ n \in \mathbb{N} $‎

‎and $\{\varphi_n\}_{n=0}^\infty$ is the inner higher derivation on $ (\mathcal{M},\pi_\textbf{e}) $ defined by‎

‎\[ \varphi_n(x)=\sum_{i=0}^n \langle v_i,\textbf{e}\rangle \langle x,\textbf{e}\rangle w_{n-i} \]‎

‎for all $ x \in \mathcal{M} $ and each non-negative integer $ n $‎,

‎then the corresponding sequence of inner derivations $\{\psi_n\}_{n=1}^\infty$‎, ‎defined on $ (\mathcal{M},\pi_\textbf{e}) $ by‎

‎\begin{align*}‎

‎\psi_n(x)&=\Big\langle \sum_{i=1}^n\Big(\sum_{\sum_{j=1}^i r_j=n}(-1)^{i-1}~r_1\langle v_{r_1},\textbf{e}\rangle \langle v_{r_2},\textbf{e}\rangle\ldots \langle v_{r_{i-1}},\textbf{e}\rangle v_{r_{i}}\Big),\textbf{e}\Big\rangle x\\‎

‎&+\langle x,\textbf{e} \rangle \sum_{i=1}^n\Big(\sum_{\sum_{j=1}^i r_j=n}(-1)^{i-1}~r_i\langle w_{r_1},\textbf{e}\rangle \langle w_{r_2},\textbf{e}\rangle \ldots \langle w_{r_{i-1}},\textbf{e}\rangle w_{r_i}\Big)\nonumber‎

‎\end{align*}‎

‎for all $ x \in \mathcal{M} $ and all $ n \in \mathbb{N} $‎.