数学学报 2011, 54(2) 333-342 DOI:      ISSN: 0583-1431 CN: 11-2038/O1

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本文关键词相关文章
Kadison--Singer格
Kadison--Singer代数
矩阵代数
本文作者相关文章
董瑷菊
侯成军
谭君
矩阵代数的Kadison--Singer格的分类
董瑷菊1, 侯成军2, 谭君3
1. 西安文理学院数学系 西安 710065;
2. 曲阜师范大学运筹所 日照 276826;
3. 中国科学院数学与系统科学研究院 北京 100190
摘要
研究了矩阵代数Mn(C)的KS格, 证明了每个生成M3(C)的KS格都相似于L0I-L0, 其中L0M3(C)的一个极大对角投影套和一个赋值全非零的秩 1 投影所生成的KS格, 从而M3(C)的对角平凡的KS代数都是$4$维的. 同时,还给出了几个生成M4(C)但非同构的KS格的例子.

 

关键词 Kadison--Singer格   Kadison--Singer代数   矩阵代数  
MSC2000 O177.1
Classification of Kadison-Singer Lattices in Matrix Algebras
Ai Ju DONG1, Cheng Jun HOU2, Jun TAN3
1. Department of Mathematics, Xi’an University of Arts and Science, Xi’an 710065, P. R. China;
2. Institute of Operations Research, Qufu Normal University, Rizhao 276826, P. R. China;
3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P. R. China
Abstract:
We study Kadison-Singer lattices in the matrix algebra Mn(C), and prove that each Kadison-Singer lattice generating M3(C) as an algebra is similar to L0 or I-L0, where L0 is the KS lattice generated by a maximal nest of diagonal projections and a rank one projection matrix with nonzero entries in M3(C), hence each Kadison-Singer algebra with trivial diagonal in M3(C) has dimension 4. In addition, we give some examples of nonisomorphic Kadison-Singer lattices which generate M4(C).

 

Keywords: Kadison-Singer algebra   Kadison-Singer lattice   matrix algebra  
收稿日期 2010-08-30 修回日期 2010-12-07 网络版发布日期  
DOI:
基金项目:

国家自然科学基金资助项目(10971117);山东省自然科学基金(ZR2009AQ005)

通讯作者:
作者简介:
作者Email: daj1965@163.com; cjhou@mail.qfnu.edu.cn

参考文献:


[1] Kadison R., Singer I., Triangular operator algebras, Fundamentals and hyper-reducible theory, Amer Journal of Math., 1960, 82: 227-259.

[2] Ge L., Yuan W., Kadison-Singer algebras, I: hyperfinite case, Proc. Natl. Acad. Sci. USA, 2010, 107(5): 1838-1843.

[3] Ge L., Yuan W., Kadison-Singer algebras, II: General case, Proc. Natl. Acad. Sci. USA, 2010, 107(11): 4840-4844.

[4] Davidson K. B., Nest Algebras, Longman Scientific & Technical, π Pitman Research Notes, New York: Mathematics Series, 1988, 191.

[5] Ringrose J., On some algebras of operators, II, Proc. London Math. Soc., 1966, 16(3): 385-402.

[6] Hou C. J., Cohomology of a class of Kadison-Singer algebras, Science China Series Mathematics, 2010, 53: 1827-1839.

[7] Hou C. J., Yuan W., Kadison-Singer lattices in finite von Neumann algebras, Preprint.

[8] Wang L. G., Yuan W., On a new class of Kadison-Singer algebras, Exposition. Math., doi:10.1016/j.exmath. 2010.08.001.

[9] Dong A. J., On triangular algebras with noncommutative diagonals, Science in China Series A: Mathematics, 2008, 51: 1937-1944.

[10] Kadison R., Ringrose J., Fundamentals of the Operator Algebras, vols. I and II, Orlando: Academic Press, 1983 and 1986.

[11] Halmos P., Reflexive lattices of subspaces, Journal of London Math. Society, 1971, 4: 257-263.

[12] Arveson W. B., Operator algebras and invariant subspaces, Ann. of Math., 1974, 100: 433-532.

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