Non-global nonlinear lie n-derivations on unital algebras with idempotents

Abstract

Let T be a unital algebra with nontrivial idempotents. For any s1, s2,... , sn ? T, define p1(s1) = s1, p2(s1, s2) = [s1, s2] and pn(s1, s2,..., sn) = [pn?1(s1, s2,..., sn?1), sn] for all integers n ? 3. In the present article, it is shown that if a map ? : T ? T satisfies ?(pn(s1, s2,..., sn)) = ?n i=1 pn(s1,..., si?1,?(si), si+1,..., sn) (n ? 3) for all s1, s2,..., sn ? T with s1s2...sn = 0, then ?(s + t) ? ?(s) ? ?(t) ? Z(T) for all s, t ? T, and under some mild assumptions ? is of the form ? + ?, where ? : T ? T is an additive derivation and ? : T ? Z(T) is a map such that ?(pn(s1, s2,..., sn)) = 0 for all s1, s2,..., sn ? T with s1s2... sn = 0. The above results are then applied to certain special classes of unital algebras, namely triangular algebras, full matrix algebras and algebra of all bounded linear operators.