Multiplicative *-Lie type higher derivations of standard operator algebras.

Let A be a standard operator algebra on an infinite dimensional complex Hilbert space H containing identity operator I. Let p n (X 1 , X 2 , ⋯ , X n) be the polynomial defined by n indeterminates X 1 , ⋯ , X n and their multiple *-Lie products and N be the set of non-negative integers. In this paper, it is shown that if A is closed under the adjoint operation and D = { d m } m ∈ N is the family of mappings d m : A → B (H) such that d 0 = i d A , the identity map on A satisfying d m (p n (U 1 , U 2 , ⋯ , U n)) = ∑ i 1 + i 2 + ⋯ + i n = m p n (d i 1 (U 1) , d i 2 (U 2) , ⋯ , d i n (U n)) for all U 1 , U 2 , ⋯ , U n ∈ A and for each m ∈ N , then D = { d m } m ∈ N is an additive *-higher derivation. Moreover, D is inner. [ABSTRACT FROM AUTHOR]