Abstract
A linear mapping T on a JB*-triple E is called triple derivable at orthogonal pairs if for every a, b, c is an element of E with a perpendicular to 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 -> A**, a central element xi is an element of A**(sa), and an anti-symmetric element eta in the multiplier algebra of A, such that
T (a) = D(a) + xi circle a + eta circle a, for all a is an element of A;
(d) There exist a triple derivation delta : A -> 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 -> M and an operator S in the centroid of M such that T = delta + S.