因子von Neumann代数上导子的等价刻画

郭玉琴, 安润玲

数学学报 ›› 2018, Vol. 61 ›› Issue (4) : 631-640.

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PDF(430 KB)
数学学报 ›› 2018, Vol. 61 ›› Issue (4) : 631-640. DOI: 10.12386/A2018sxxb0058
论文

因子von Neumann代数上导子的等价刻画

    郭玉琴, 安润玲
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Equivalent Characterization of Derivations on Factor von Neumann Algebras

    Yu Qin GUO, Run Ling AN
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摘要

R是含非平凡幂等元P的素环,CRC=PC.本文证明可加映射△:RRC可导,即△(AB)=△(AB+A△(B),∀ A,B∈RAB=C当且仅当存在导子δ:{R}→{R},使得△(A)=δA)+△(I)A,∀ AR.没有I1型中心直和项的von Neumann代数上的可导映射也有类似结论.利用该结论证明了,若非零算子CBX),使得ran(C或ker(C)在X中可补,则可加映射△:BX)→BX)在C可导当且仅当它是导子.特别地,证明了因子von Neumann代数上的可加映射在任意但固定的非零算子可导当且仅当它是导子.

Abstract

Let R be a prime ring containing a nontrivial idempotent P. Suppose CR satisfies C=PC. It is shown that an additive map Δ:RR is derivable at C, that is, Δ(AB)=Δ(A)B + AΔ(B) for every A, BR with AB=C if and only if there exists a derivation δ:RR such that Δ(A)=δ(A) + Δ(I)A for all AR. Similar results are obtained for von Neumann algebras with no central abelian projections. As its application, we obtain that, if nonzero operator CB(X) such that ran(C) or ker(C) is complementary in X, then an additive map Δ:B(X) → B(X) is derivable at C if and only if it is a derivation. In particular, we show that an additive map from a factor von Neumann algebra into itself is derivable at an arbitrary but fixed nonzero operator if and only if it is a derivation.

关键词

素环 / 导子 / 可补子空间 / von Neumann代数 / 中心覆盖

Key words

prime rings / derivations / complementary subspaces / von Neumann algebras / central carrier

引用本文

导出引用
郭玉琴, 安润玲. 因子von Neumann代数上导子的等价刻画. 数学学报, 2018, 61(4): 631-640 https://doi.org/10.12386/A2018sxxb0058
Yu Qin GUO, Run Ling AN. Equivalent Characterization of Derivations on Factor von Neumann Algebras. Acta Mathematica Sinica, Chinese Series, 2018, 61(4): 631-640 https://doi.org/10.12386/A2018sxxb0058

参考文献

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基金

国家自然科学基金资助项目(11001194)

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