Kadison-Singer代数研究综述

董瑷菊, 陈广锋, 杨渭清, 张运良, 安军龙

数学学报 ›› 2018, Vol. 61 ›› Issue (2) : 177-196.

PDF(794 KB)
PDF(794 KB)
数学学报 ›› 2018, Vol. 61 ›› Issue (2) : 177-196. DOI: 10.12386/A2018sxxb0017
论文

Kadison-Singer代数研究综述

    董瑷菊, 陈广锋, 杨渭清, 张运良, 安军龙
作者信息 +

Survey on Kadison-Singer Algebras

    Ai Ju DONG, Guang Feng CHEN, Wei Qing YANG, Yun Liang ZHANG, Jun Long AN
Author information +
文章历史 +

摘要

回顾了建立KS-代数的研究背景,系统介绍了KS-代数的定义和性质以及超有限KS-代数、非超有限KS-代数、KS-格的构造和强KS-代数的研究结果,同时分析了KS-代数和经典的不变子空间、Kadison可迁代数、von Neumann代数生成元等问题之间的联系;讨论了非自伴代数的运算,给出了两种不同构造非自伴代数的运算法则;在此基础上,提出了未来学科发展有待研究的16个问题.

Abstract

The background for the introduction of Kadison-Singer algebras (KSalgebras, for short) is reviewed. Definitions and basics properties are explained. Studies on hyperfinite KS-algebras, non-hyperfinite case, KS-lattices and strong KS-algebras are described in details, as well as their connections with classical open problems such as the invariant subspace problem, Kadison's transitive algebra problem, von Neumann algebra generator problem. Operations on non-selfadjoint operator algebras are also discussed and two new operations are included in the discussion. Sixteen open problems are listed with some explanations in the end.

关键词

自反代数 / 自反子空间格 / von Neumann代数 / KS-代数 / 强KS-代数

Key words

reflexive algebras / reflexive lattices / von Neumann algebras / KS-algebras / strong KS-algberas

引用本文

导出引用
董瑷菊, 陈广锋, 杨渭清, 张运良, 安军龙. Kadison-Singer代数研究综述. 数学学报, 2018, 61(2): 177-196 https://doi.org/10.12386/A2018sxxb0017
Ai Ju DONG, Guang Feng CHEN, Wei Qing YANG, Yun Liang ZHANG, Jun Long AN. Survey on Kadison-Singer Algebras. Acta Mathematica Sinica, Chinese Series, 2018, 61(2): 177-196 https://doi.org/10.12386/A2018sxxb0017

参考文献

[1] Davidson K. R., Nest Algebras, Longman Scientific & Technical, π Pitman Research Notes in Mathematics Series, Vol.191, New York, 1988.
[2] Dong A., On triangular algebras with noncommutative diagonals, Science in China, Ser. A, 2008, 51:1937-1944.
[3] Dong A., Skew products of operator algebras, Acta Math. Sinica, Chinese Ser., 2016, 59:639-644.
[4] Dong A., Hou C., Yuan W., et al., On representations and operations of reflexive lattices (in Chinese), Scientia Sinica Math., 2012, 42:321-328.
[5] Dong A., Hou C., On some maximal non-selfadjoint operator algebras, Expositiones Mathematicae, 2012, 30:309-317.
[6] Dong A., Hou C., On some automorphisms of a class of Kadison-Singer algebras, Linear Algebra and Its Applications, 2012, 436:2037-2053.
[7] Dong A., Hou C., Tan J., Classification of Kadison-Singer lattices in matrix algebras, Acta Math. Sinica, Chinese Ser., 2011, 54:333-342.
[8] Dong A., Wang D., On strong Kadison-Singer algebras, Acta Math. Sinica, Engl., Ser., 2013, 29:2219-2232.
[9] Dong A., Wang D., On maximal non-selfadjoint reflexive algebras associated with a double triangle lattice, Science China Math., 2015, 58:2373-2386.
[10] Enflo P., On the invariant subspace problem for Banach spaces, Acta Math., 1987, 158:213-313.
[11] Ge L., Problems in Operator Algebras, Talks in 2009 Shanghai and in 2015 Xi'an.
[12] Ge L., Popa S., On some decomposition properties for factors of type Ⅱ1, Duke Math. J., 1998, 94:79-101.
[13] Ge L., Yuan W., Kadison-Singer algebras, I:hyperfinite case, Proc. Natl. Acad. Sci. U.S.A., 2010, 107:1838-1843.
[14] Ge L., Yuan W., Kadison-Singer algebras, Ⅱ:general case, Proc. Natl. Acad. Sci. U.S.A., 2010, 107:4840-4844.
[15] Halmos P., Reflexive lattices of subspaces, J. London Math. Soc., 1971, 4:257-263.
[16] Hou C., Cohomology of a class of Kadison-Singer algebras, Science China:Mathematics, 2010, 53:1827-1839.
[17] Hou C., Zhang H., A note on the diagonal maximality of operator algebras, Linear Algebra and Its Applications, 2012, 436:2406-2418.
[18] Hou C., Yuan W., Minimal generating reflexive lattices of projections in finite von Neumann algebras, Math. Ann., 2012, 353:499-517.
[19] Kadison R., On the orthogonalization of operator representations, Amer. J. Math., 1955, 78:600-621.
[20] Kadison R., Ringrose J., Fundamentals of the Operator Algebras, vols. I and Ⅱ, Academic Press (Orlando), 1983 and 1986.
[21] Kadison R., Singer I., Triangular operator algebras, Amer. J. Math., 1960, 82:227-259.
[22] Kadison R., Singer I., Triangular operator algebras, Another chapter, Contemp Math., 1991, 120:63-76.
[23] Murray F., von Neumann J., Rings of operators, Ann. Math., 1936, 37:116-229.
[24] Orr J., On the closure of triangular algebras, Amer. J. Math., 1990, 112:481-497.
[25] Radjavi H., Rosenthal P., Invariant Subspaces, Springer-Verlag, Berlin, 1973.
[26] Ravichandran M., Kadison-Singer algebras with applications to von Neumann algebras, Ph.D. dissertation, Durham:University of New Hampshire, 2009.
[27] Ringrose J., On some algebras of operators, Proc. London Math. Soc., 1965, 15:61-83.
[28] Ren Y., Wu W., Some New Classes of Kadison-Singer Lattices in Hilbert Spaces, Science China, Math., 2014, 57:837-846.
[29] Wang L., Yuan W., A new class of Kadison-Singer algebras, Expositiones Mathematicae, 2011, 29:126-132.
[30] Wu W., Yuan W., On generators of abelian Kadison-Singer algebras in matrix algebras, Linear Algebra App., 2014, 440:197-205.

基金

国家自然科学基金资助项目(11371290);陕西自然科学基础研究计划项目(2017JM1045)

PDF(794 KB)

Accesses

Citation

Detail

段落导航
相关文章

/