Multiplicative *-Lordan 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 p_n (X ₁ , X ₂ , · · ·, X_n ) be the polynomial defined by n indeterminates X ₁, · · ·, X_n and their Jordan multiple ∗-products. In this paper it is shown that a family ? = {d_m } _m ∈ℕ of mappings  such that , the identity map on  satisfies the condition

for all U ₁ , U ₂ , · · ·, U_n ∈  and for each m ∈ ℕ if and only if ? = {d_m } _m ∈ℕ is an additive ∗-higher derivation.

Mathematics Subject Classification (2010): 47B47, 16W25, 46K15.