Lie triple derivations of standard operator algebras

Let X be a Banach space over the field F (R or C). Denote by B(X) the set of all bounded linear operators on X and by F(X) the set of all finite rank operators on X. A subalgebra A subset of B(X) is called a standard operator algebra if F(X) subset of A. Suppose that delta is a mapping from A into B(X). First, we prove that if delta is a Lie triple derivation, then delta is standard. Next, we show that if delta is a local Lie triple derivation and dim(X) >= 3, then delta is a Lie triple derivation. Finally, we prove that if delta is a 2-local Lie triple derivation, then delta = d + tau, where d is a derivation, and tau is a homogeneous mapping from A into F/ such that tau(A + B) = tau (A) for each A, B in A where B is a sum of double commutators.

Automated summary

This article discusses properties and definitions of operator algebras and derivations. It proves the standardness of Lie triple derivations and the relationship between local Lie triple derivations and Lie triple derivations. Furthermore, it provides the explicit form of 2-local Lie triple derivations.