Multiplicative ∗-lie triple higher derivations of standard operator algebras

Abstract

Let A be a standard operator algebra on an infinite dimensional complex Hilbert space H containing identity operator I. In this paper it is shown that if A is closed under the adjoint operation, then every multiplicative ∗-Lie triple derivation d : A → B(H) is a linear ∗-derivation. Moreover, if there exists an operator S ∈ B(H) such that S + S ∗ = 0 then d(U) = US − SU for all U ∈ A, that is, d is inner. Furthermore, it is also shown that any multiplicative ∗-Lie triple higher derivation D = {δn}n∈N of A is automatically a linear inner higher derivation on A with d(U) ∗ = d(U ∗ ).

Key words: Multiplicative ∗-Lie derivation, multiplicative ∗-Lie triple higher derivation, standard operator algebra.