In this paper, we prove that every * -Lie derivable mapping on a von Neumann algebra with no central abelian projections can be expressed as the sum of an additive * -derivation and a mapping with image in the center vanishing at commutators.
In this paper, it is shown that if A is a CSL subalgebra of a von Neumann algebr and φ is a continuous mapping on A such that (m + n + k + l)φ(A 2 ) − (mφ(A)A + nAφ(A) + kφ(I)A 2 + lA 2 φ(I)) ∈ FI for any A ∈ A, where F is the real field or the complex field, then φ is a centralizer. It is also shown that if φ is an additive mapping on A such that (m + n + k + l)φ(A 2 ) = mφ(A)A + nAφ(A) + kφ(I)A 2 + lA 2 φ(I) for any A ∈ A, then φ is a centralizer.
In this paper, it is shown that if T=Tri(A,M,B) is a triangular algebra and ϕ is an additive operator on T such that (m+n+k+l)ϕ(T2)-(mϕ(T)T+nTϕ(T)+kϕ(I)T2+lT2ϕ(I))∈FI for any T∈T, then ϕ is a centralizer. It follows that an (m,n)- Jordan centralizer on a triangular algebra is a centralizer.
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