Multiplicative ∗-Jordan type higher derivations on von Neumann algebras

Abstract

Let be a von Neumann algebra without nonzero central abelian projections on a complex Hilbert space . Let pn (X 1 , X 2 , · · ·, Xn ) be the polynomial defined by n indeterminates X 1, · · ·, Xn and their Jordan multiple ∗-products. In this paper it is shown that a family 𝒟 = {dm } m ∈ℕ of mappings such that , the identity map on satisfies the condition for all U 1 , U 2 , · · ·, Un ∈ and for each m ∈ ℕ if and only if 𝒟 = {dm } m ∈ℕ is an additive ∗-higher derivation.