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

Abstract

A linear mapping $T$ on a JB$^*$-triple is called triple derivable at orthogonal pairs if for every $a,b,c\in E$ with $a\perp b$ we have $$0 = \{T(a), b,c\} + \{a,T(b),c\}+\{a,b,T(c)\}.$$ 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 $D:A\to A^{**}$, a central element $\xi\in A^{**}_{sa},$ and an anti-symmetric element $\eta$ in the multiplier algebra of $A$, such that $$ T(a) = D(a) + \xi \circ a + \eta \circ a, \hbox{ for all } a\in A;$$ $(d)$ There exist a triple derivation $\delta: A\to A^{**}$ 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 $\delta: M\to M$ and an operator $S$ in the centroid of $M$ such that $T = \delta + S$. \end{enumerate}