矩阵代数的Kadison--Singer格的分类

董瑷菊, 侯成军, 谭君

数学学报 ›› 2011, Vol. 54 ›› Issue (2) : 333-342.

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数学学报 ›› 2011, Vol. 54 ›› Issue (2) : 333-342. DOI: 10.12386/A2011sxxb0035
论文

矩阵代数的Kadison--Singer格的分类

    董瑷菊1, 侯成军2, 谭君3
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Classification of Kadison-Singer Lattices in Matrix Algebras

    Ai Ju DONG1, Cheng Jun HOU2, Jun TAN3
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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.

 

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).

 

关键词

Kadison--Singer格 / Kadison--Singer代数 / 矩阵代数

Key words

Kadison-Singer algebra / Kadison-Singer lattice / matrix algebra

引用本文

导出引用
董瑷菊, 侯成军, 谭君. 矩阵代数的Kadison--Singer格的分类. 数学学报, 2011, 54(2): 333-342 https://doi.org/10.12386/A2011sxxb0035
Ai Ju DONG, Cheng Jun HOU, Jun TAN. Classification of Kadison-Singer Lattices in Matrix Algebras. Acta Mathematica Sinica, Chinese Series, 2011, 54(2): 333-342 https://doi.org/10.12386/A2011sxxb0035

参考文献


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[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.

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基金

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

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