von Neumann代数上的局部可乘Lie n-导子

齐霄霏, 冯小雪

数学学报 ›› 2020, Vol. 63 ›› Issue (4) : 349-366.

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数学学报 ›› 2020, Vol. 63 ›› Issue (4) : 349-366. DOI: 10.12386/A2020sxxb0030
论文

von Neumann代数上的局部可乘Lie n-导子

    齐霄霏, 冯小雪
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Local Multiplicative Lie n-derivations on von Neumann Algebras

    Xiao Fei QI, Xiao Xue FENG
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文章历史 +

摘要

A1,…,An的(n-1)-换位子记为pnA1,…,An).令M是von Neumann代数,n ≥ 2是任意正整数,LMM是一个映射.本文证明了,若M不含I1型中心直和项,且L满足LpnA1,…,An))=∑k=1n pnA1,…,Ak-1LAk),Ak+1,…,An)对所有满足条件A1A2=0的A1A2,…,AnM成立,则LA)=φA)+fA)对所有AM成立,其中φMMfMZM)(M的中心)是两个映射,且满足φPiMPj上是可加导子,fpnA1A2,…,An))=0对所有满足A1A2=0的A1A2,…,AnPiMPj成立(1 ≤ i,j ≤ 2),P1M是core-free投影,P2=I-P1;若M还是因子且n ≥ 3,则L满足条件LpnA1A2,…,An))=∑k=1n pnA1,…,Ak-1LAk),Ak+1,…,An)对所有满足A1A2A1=0的A1A2,…,AnM成立当且仅当LA)=φA)+hAI对所有AM成立,其中φM上的可加导子,hM上的泛函且满足hpnA1A2,…,An))=0对所有满足条件A1A2A1=0的A1A2,…,AnM成立.

Abstract

Denote by pn(A1, …, An) the (n-1)-commutator of A1, …, An. Assume that M is a von Neumann algebra, n ≥ 2 is any positive integer and L:MM is a mapping. It is shown that, if M has no central summands of type I1 and L satisfies L(pn(A1, …, An))=∑k=1n pn(A1, …, Ak-1, L(Ak), Ak+1, …, An) for all A1, A2, …, AnM with A1A2=0, then L(A)=φ(A) + f(A) for all AM, where φ:MM and f:MZ (M) (the center of M) are two mappings such that the restriction to PiMPj of φ is an additive derivation and f(pn(A1, A2, …, An))=0 for all A1, A2, …, AnPiMPj with A1A2=0 (1 ≤ i, j ≤ 2), P1M is a core-free projection and P2=I -P1; if M is a factor and n ≥ 3, then L satisfies L(pn(A1, A2, …, An))=∑k=1n pn(A1, …, Ak-1, L(Ak), Ak+1, …, An) for all A1, A2, …, AnM with A1A2A1=0 if and only if L(A)=φ(A) + h(A)I for all AM, where φ is an additive derivation on M and h is a functional of M such that h(pn(A1, A2, …, An))=0 for all A1, A2, …, AnM with A1A2A1=0.

关键词

von Neumann代数 / Lie n-导子 / Lie导子 / Lie triple导子

Key words

von Neumann algebras / Lie n-derivations / Lie derivations / Lie triple derivations

引用本文

导出引用
齐霄霏, 冯小雪. von Neumann代数上的局部可乘Lie n-导子. 数学学报, 2020, 63(4): 349-366 https://doi.org/10.12386/A2020sxxb0030
Xiao Fei QI, Xiao Xue FENG. Local Multiplicative Lie n-derivations on von Neumann Algebras. Acta Mathematica Sinica, Chinese Series, 2020, 63(4): 349-366 https://doi.org/10.12386/A2020sxxb0030

参考文献

[1] Abdullaev I. Z., n-Lie derivations on von Neumann algebras. Uzbek. Mat. Zh., 1992, 5-6:3-9.
[2] Bai Z. F., Du S. P., The structure of nonlinear Lie derivation on von Neumann algebras. Linear Algebra Appl., 2012, 436:2701-2708.
[3] Benkovi? D., Eremita D., Multiplicative Lie n-derivations of triangular rings, Linear Algebra Appl., 2012, 436:4223-4240.
[4] Cheung W. S., Lie derivations of triangular algebras. Linear and Multilinear Algebra, 51, 299-310.
[5] Fošner A., Wei F., Xiao Z. K., Nonlinear Lie-type derivations of von Neumann algebras and related topics, Colloq. Math., 2013, 132:53-71.
[6] Halmos P., A Hilbert Space Problem Book, 2nd Edition, Springer-Verlag, New York, 1982.
[7] Kadison R. V., Ringrose J. R., Fundamentals of the Theory of Operator Algebras, Vol. I, Academic Press, New York, 1983; Vol. II, Academic Press, New York, 1986.
[8] Kleinecke D. C., On operator commutators, Proc. Amer. Math. Soc., 1957, 50:138-142.
[9] Liu L., Lie triple derivations on factor von Neumann algebras, Bull. Korean Math. Soc., 2015, 52:581-591.
[10] Lu F. Y., Liu B. H., Lie derivations of reflexive algebras, Integr. Equ. Oper. Theory, 2009, 64:261-271.
[11] Mathieu M., Villena A. R., The structure of Lie derivations on C*-algebras, J. Funct. Anal., 2003, 202:504-525.
[12] Miers C. R., Lie triple derivations of von Neumann algebras, Proc. Amer. Math. Soc., 1978, 71:57-61.
[13] Qi X. F., Characterizing Lie n-derivations for reflexive algebras, Linear and Multilinear Algebra, 2015, 63:1693-1706.
[14] Shirokov F. V., Proof of a conjecture of Kaplansky, Uspehi Mat. Nauk, 1956, 11:167-168.
[15] Wang Y., Wang Y., Multiplicative Lie n-derivations of generalized matrix algebras, Linear Algebra Appl., 2013, 438:2599-2616.

基金

国家自然科学基金资助项目(11671006);山西省优秀青年基金资助项目(201701D211001)

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