Higher Lie n-Centralizers and Generalized Higher Lie n-Derivations of von Neumann Algebras

Let U be a von Neumann algebra, n≥2, P∈U a projection, and let p1(A1)=A1 and pn(A1,…,An)=[pn−1(A1,…,An−1),An]. Fix a higher Lie n-derivation Δ={δm}m=0N of U with δ0=id, of rank N∈N∪{∞}. We classify the families Φ={φm}m=0N of additive maps satisfying φm(pn(A1,…,An))=∑i1+…+in=mpn(φi1(A1),δi2(A2),…,δin(An))=∑i1+…+in=mpn(δi1(A1),φi2(A2),…,δin(An)) for all A1,…,An∈U with A1A2=P. In terms of the generating series Φ(t)=∑mφmtm and Δ(t)=∑mδmtm, these are exactly the families Φ(t)=W(t)Δ(t)+Ξ(t) with W(t) a power series over the center of U and Ξ(t) a series of center-valued maps annihilating the pn(A1,…,An) with A1A2=P. If U has no central summands of type I1, then W(t) and Ξ(t) are unique, each ξm vanishes on PU(I−P) and on (I−P)UP, and the solution set splits as the free rank-one module on Δ plus the null families. The underlying reduction of the rank-N problem to the rank-zero problem is valid over any ring, and we apply it also to upper triangular 2×2 matrices over a commutative ring. Moreover, generalized higher Lie n-derivations are characterized as a consequence.