Characterization of nonlinear Lie type derivations on von Neumann algebras by local action

Abstract

Let R be a unital ring containing a nontrivial idempotent. In this article, under a mild condition on R , we prove that if a map δ : R → R satisfies δ ( P n ( A 1 , A 2 , A 3 , … , A n ) ) = ∑ i = 1 n P n ( A 1 , … , A i − 1 , δ ( A i ) , A i + 1 , … , A n ) for any A 1 , A 2 , A 3 , … , A n ∈ R with A 1 A 2 A 3 = 0 , then δ ( A + B ) − δ ( A ) − δ ( B ) ∈ Z ( R ) for any A , B ∈ R . In particular, if R is a von Neumann algebra with no central summands of type I 1 or a factor, then δ ( x ) = d ( x ) + τ ( x ) for all x ∈ R , where d : R → R is an additive derivation and τ : R → Z ( R ) is a map vanishing on each ( n − 1 ) -th commutator p n ( A 1 , A 2 , A 3 , … , A n ) with A 1 A 2 A 3 = 0 .