A linear mapping T on a JB\(^*\)-triple E is called triple derivable at orthogonal pairs if for every \(a,b,c\in E\) with \(a\perp b\) we have

We prove that for each bounded linear mapping T on a JB\(^*\)-algebra A the following assertions are equivalent:

(a):

T is triple derivable at zero;

(b):

T is triple derivable at orthogonal elements;

(c):

There exists a Jordan \(^*\)-derivation \(D:A\rightarrow A^{**}\), a central element \(\xi \in A^{**}_{sa},\) and an anti-symmetric element \(\eta \) in the multiplier algebra of A, such that

$$\begin{aligned} T(a) = D(a) + \xi \circ a + \eta \circ a, \hbox { for all } a\in A; \end{aligned}$$

(d):

There exist a triple derivation \(\delta : A\rightarrow A^{**}\) and a symmetric element S in the centroid of \(A^{**}\) such that \(T= \delta +S\).

The result is new even in the case of C\(^*\)-algebras. We next establish a new characterization of those linear maps on a JBW\(^*\)-triple which are triple derivations in terms of a good local behavior on Peirce 2-subspaces. We also prove that assuming some extra conditions on a JBW\(^*\)-triple M, the following statements are equivalent for each bounded linear mapping T on M:

(a):

T is triple derivable at orthogonal pairs;

(b):

There exists a triple derivation \(\delta : M\rightarrow M\) and an operator S in the centroid of M such that \(T = \delta + S\).

JB \(^*\) -triple E上的线性映射T称为三重可在正交对上可推导，如果对于每个\(a,b,c\in E\)和\(a\perp b\)我们有

我们证明，对于JB \(^*\) -代数A上的每个有界线性映射T，以下断言是等效的：

（一种）：

T是可在零处三重推导的；

(二)：

T在正交元素处是三重可推导的；

（C）：

存在一个 Jordan \(^*\) -派生\(D:A\rightarrow A^{**}\)，一个中心元素\(\xi \in A^{**}_{sa},\)和A的乘数代数中的反对称元素\(\eta \)，使得

$$\begin{aligned} T(a) = D(a) + \xi \circ a + \eta \circ a, \hbox { for all } a\in A; \end{对齐}$$

(四)：

存在三重推导\（\增量：A \ RIGHTARROW甲^ {**} \）和对称元件š中的质心\（A ^ {**} \） ，使得\（T = \增量+ S \)。

即使在 C \(^*\) -代数的情况下，结果也是新的。我们接下来在JBW \(^*\) -triple上建立这些线性映射的新特征，这些线性映射是根据 Peirce 2-子空间上的良好局部行为的三重推导。我们还证明，假设JBW \(^*\) -triple M上的一些额外条件，以下语句对于M上的每个有界线性映射T是等效的：

（一种）：

T在正交对上是三重可推导的；

(二)：

存在三重推导\（\增量：M \ RIGHTARROW中号\）和操作者小号中的质心中号，使得\（T = \增量+ S \） 。