Characterizing Derivations on Von Neumann Algebras by Local Actions

Xiaofei Qi,Jia Ji

doi:10.1155/2013/407427

Abstract

Letℳbe any von Neumann algebra without central summands of typeI1andPa core-free projection with the central carrierI. For any scalarξ, it is shown that every additive mapLonℳsatisfiesL(AB-ξBA)=L(A)B-ξBL(A)+AL(B)-ξL(B)AwheneverAB=Pif and only if (1)ξ=1,L=φ+h, whereφis an additive derivation andhis a central valued additive map vanishing onAB-BAwithAB=P; (2)ξ≠1,Lis a derivation withL(ξA)=ξL(A)for eachA∈ℳ.

Highlights

Let A be an algebra over a field F
The purpose of the present paper is to give a complete characterization of additive maps L satisfying L([A, B]ξ) = [L(A), B]ξ+[A, L(B)]ξ for any A, B with AB = P on von Neumann algebras without central summands of type I1 for all possible ξ
We prove that L satisfies [L(A), B]ξ + [A, L(B)]ξ = L([A, B]ξ) for any A, B ∈ M with AB = P if and only if (1) ξ = 1, L(A) = φ(A)+h(A) for all A ∈ M, where φ : M → M is an additive derivation, h : M → Z(M) is an additive map vanishing each commutator [A, B] whenever AB = P, and Z(M) denotes the center of M (Theorem 1); (2) ξ ≠ 1, L is an additive derivation satisfying L(ξA) = ξL(A) for all A (Theorem 4)

Summary

Introduction

Let A be an algebra over a field F. By [17], if M is a von Neumann algebra without central summands of type I1, there exists a nonzero core-free projection P ∈ M with P = I. The purpose of the present paper is to give a complete characterization of additive maps L satisfying L([A, B]ξ) = [L(A), B]ξ+[A, L(B)]ξ for any A, B with AB = P on von Neumann algebras without central summands of type I1 for all possible ξ. Let M be a von Neumann algebra without central summands of type I1 and P ∈ M a nonzero core-free projection with P = I.

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