2020/09/22 by Essaleh, Ahlem Ben Ali, Peralta, Antonio M.
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2009.10336
A linear mapping T on a JB^*-triple is called triple derivable at orthogonal pairs if for every a,b,c∈ E with a⊥ 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 ξ∈ A**sa, and an anti-symmetric element η in the multiplier algebra of A, such that T(a) = D(a) + ξ∘ a + η∘ a, \hbox for all a∈ A; (d) There exist a triple derivation δ: A→ A** and a symmetric element S in the centroid of A** such that T= δ+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 δ: M→ M and an operator S in the centroid of M such that T = δ+ S. \endenumerate