Turkish Journal of Mathematics




Let $\mathcal{A}$ be a unital, complex normed $\ast$-algebra with the identity element $\textbf{e}$ such that the set of all algebraic elements of $\mathcal{A}$ is norm dense in the set of all self-adjoint elements of $\mathcal{A}$ and let $\{D_n\}_{n = 0}^{\infty}$ and $\{\Delta_n\}_{n = 0}^{\infty}$ be sequences of continuous linear mappings on $\mathcal{A}$ satisfying \[ \left\lbrace \begin{array}{c l} D_{n + 1}(p) = \sum_{k = 0}^{n}D_{n - k}(p)D_k(p),\\ \\ \Delta_{n + 1}(p) = \sum_{k = 0}^{n}\Delta_{n - k}(p)D_k(p), \end{array} \right. \] for all projections $p$ of $\mathcal{A}$ and all nonnegative integers $n$. Moreover, suppose that $D_0(p) = D_0(p)^2$ holds for all projections $p$ of $\mathcal{A}$. Then \begin{align*} \Delta_n = \frac{C_n}{2}\left(R_{D_0(\textbf{e})}\Delta_0 + L_{\Delta_0(\textbf{e})}D_0 \right) \end{align*} for all $n \in \mathbb{N}$, where $C_n$ denotes the $n$th Catalan number and $R_{D_0(\textbf{e})}(a) = a D_0(\textbf{e})$ and $L_{\Delta_0(\textbf{e})}(a) = \Delta_0(\textbf{e})a$ for all $a \in \mathcal{A}$. Using this result, we present a characterization of left $\tau$-centralizers satisfying a certain recursive relation. In addition, a characterization of generalized higher derivations is presented. Moreover, we show that higher derivations, prime higher derivations, left higher derivations, and $\sigma$-derivations are zero under certain conditions.

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