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2 Colloquium Mathematicum

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

100%

De Filippis V. , Wei F.

nr 1

61-74

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.

Nonlinear Lie-type derivations of von Neumann algebras and related topics

80%

Fošner A. , Wei F. , Xiao Z.

tom 132

nr 1

53-71

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.