A characterization of derivations of JSL algebras

Abstract

Let Alg L be a J -subspace lattice algebra on a Banach space X and Inv(Alg L) be the set of all invertible elements of Alg L. Suppose that δ : Alg L →Alg L is a linear mapping satisfying δ(A) ◦ A⁻¹ + A ◦ δ(A⁻¹) = 2δ(I) with A ∈ Inv(Alg L), where I is the identity element of Alg L and ◦ denotes the Jordan product A ◦ B = AB + BA for all A, B ∈ Alg L. We show that δ is a derivation. This result can apply to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.  Moreover, we give a characterization of Jordan derivation on Banach algebra with unity by the consideration of a  continuous bilinear map.