Characterizing Derivations on Von Neumann Algebras by Local Actions

Abstract EN

Let ℳ be any von Neumann algebra without central summands of type I1 and P a core-free projection with the central carrier I.

For any scalar ξ, it is shown that every additive map L on ℳ satisfies L(AB-ξBA)=L(A)B-ξBL(A)+AL(B)-ξL(B)A whenever AB=P if and only if (1) ξ=1, L=φ+h, where φ is an additive derivation and h is a central valued additive map vanishing on AB-BA with AB=P; (2) ξ≠1, L is a derivation with L(ξA)=ξL(A) for each A∈ℳ.