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Posner's second theorem and annihilator conditions with generalized skew derivations

100%
De Filippis V., Wei F.
Colloquium Mathematicum
|
2012
|
tom 129
|
nr 1
61-74
EN
Let 𝓡 be a prime ring of characteristic different from 2, $𝒬_r$ be its right Martindale quotient ring and 𝓒 be its extended centroid. Suppose that 𝒢 is a non-zero generalized skew derivation of 𝓡 and f(x₁,..., xₙ) is a non-central multilinear polynomial over 𝓒 with n non-commuting variables. If there exists a non-zero element a of 𝓡 such that a[𝒢 (f(r₁,..., rₙ)),f(r₁, ..., rₙ)] = 0 for all r₁, ..., rₙ ∈ 𝓡, then one of the following holds: (a) there exists λ ∈ 𝓒 such that 𝒢 (x) = λx for all x ∈ 𝓡; (b) there exist $q ∈ 𝒬_r$ and λ ∈ 𝓒 such that 𝒢 (x) = (q+λ)x + xq for all x ∈ 𝓡 and f(x₁, ..., xₙ)² is central-valued on 𝓡.
2
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Lie derivations of dual extensions of algebras

100%
Li Y., Wei F.
Colloquium Mathematicum
|
2015
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tom 141
|
nr 1
65-82
EN
Let K be a field and Γ a finite quiver without oriented cycles. Let Λ := K(Γ,ρ) be the quotient algebra of the path algebra KΓ by the ideal generated by ρ, and let 𝒟(Λ) be the dual extension of Λ. We prove that each Lie derivation of 𝒟(Λ) is of the standard form.
3
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Nonlinear Lie-type derivations of von Neumann algebras and related topics

81%
Fošner A., Wei F., Xiao Z.
Colloquium Mathematicum
|
2013
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tom 132
|
nr 1
53-71
EN
Motivated by the powerful and elegant works of Miers (1971, 1973, 1978) we mainly study nonlinear Lie-type derivations of von Neumann algebras. Let 𝓐 be a von Neumann algebra without abelian central summands of type I₁. It is shown that every nonlinear Lie n-derivation of 𝓐 has the standard form, that is, can be expressed as a sum of an additive derivation and a central-valued mapping which annihilates each (n-1)th commutator of 𝓐. Several potential research topics related to our work are also presented.
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