A linear preserver problem on maps which are triple derivable at orthogonal pairs

Abstract

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

$$\begin{aligned} 0 = \{T(a), b,c\} + \{a,T(b),c\}+\{a,b,T(c)\}. \end{aligned}$$

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 $T = \delta + S$.