算子代数的斜积

董瑷菊

数学学报 ›› 2016, Vol. 59 ›› Issue (5) : 639-644.

PDF(519 KB)
PDF(519 KB)
数学学报 ›› 2016, Vol. 59 ›› Issue (5) : 639-644. DOI: 10.12386/A2016sxxb0058
论文

算子代数的斜积

    董瑷菊
作者信息 +

On Skew Product of Operator Algebras

    Ai Ju DONG
Author information +
文章历史 +

摘要

{引入了算子代数的一种新运算“斜积”,证明了在这个新定义的斜积运算下算子代数的自反性保持不变.研究发现,斜积运算对应的子空间格是拓扑意义下的格的直积关系.这个新发现的重要意义在于由此可从已知的自反子空间格生成更多更复杂的新自反格,从而得到新的自反代数.在此基础上,本文对KS-代数保持性等其他非自伴代数类的性质也作了相应研究.

Abstract

A new operation, skew product, of operator algebras is introduced. We show that reflexivity of operator algebras is preserved under the skew product. Thus many new reflexive algebras can be constructed. We also show that the skew product of two KS-algebras is, in general, not a KS-algebra.

关键词

von Neumann代数 / Kadison-Singer代数 / 自反代数

Key words

von Neumann algebras / Kadison-Singer algebras / reflexive algebras

引用本文

导出引用
董瑷菊. 算子代数的斜积. 数学学报, 2016, 59(5): 639-644 https://doi.org/10.12386/A2016sxxb0058
Ai Ju DONG. On Skew Product of Operator Algebras. Acta Mathematica Sinica, Chinese Series, 2016, 59(5): 639-644 https://doi.org/10.12386/A2016sxxb0058

参考文献

[1] Davidson K., Nest Algebras, Longman Scientific & Technical, π Pitman Research Notes in Mathematics Series, 191, New York:Longman, 1988.
[2] Dong A., On triangular algebras with noncommutative diagonals (in Chinese), Sci. Sin. Math., 2008, 38:897-903.
[3] Dong A., Hou C., Yuan W., Chen G., Representations and operations on reflexive subspace lattices (in Chinese), Sci. Sin. Math., 2012, 42:321-328.
[4] Ge L., On "Problems on von Neumann algebras by R Kadison, 1967", Acta Mathematica Sinica, 2003, 19:619-624.
[5] Ge L., Yuan W., Kadison-Singer algebras, I:Hyperfinite case., Proc. Natl. Acad. Sci. USA, 2010, 107:1838-1843.
[6] Ge L., Yuan W., Kadison-Singer algebras, Ⅱ:General case., Proc. Natl. Acad. Sci. USA, 2010, 107:4840-4844.
[7] Hou C., Yuan W., Kadison-Singer lattices in finite von Neumann algebras, Math. Ann., 2011, DOI:10.1007/s00208-011-0695-7.
[8] Kadison R., Ringrose J., Fundamentals of the Operator Algebras, vols, I and Ⅱ, Orlando:Academic Press, 1983 and 1986.
[9] Kadison R., Singer I., Triangular operator algebras, fundamentals and hyper-reducible theory, Amer. J. Math., 1960, 82:227-259.
[10] Kadison R., Singer I., Triangular operator algebras Another chapter, In:Contemp Math., 120, Providence, RI:Amer. Math. Soc., 1991:63-76.
[11] von Neumann J., Rings of operators, Math. Ann., 1930, 102:370-427.
[12] Ravichandran M., Kadison-Singer algebras with applications to von Neumann algebras, Ph.D dissertation, University of New Hampshire, Durham, 2009.
[13] Ren Y., Wu W., Some new classes of Kadison-Singer lattices in Hilbert spaces, Sci. China, Math., 2014, 57:837-846.
[14] Ringrose J., On some algebras of operators, Ⅱ, Proc. London. Math. Soc., 1966, 16:385-402.
[15] Voiculescu D., Dykema K., Nica A., "Free Random Variables", CRM Monograph Series, Vol.1, AMS, Providence, R.I., 1992.

基金

国家自然科学基金资助项目(11371290)

PDF(519 KB)

Accesses

Citation

Detail

段落导航
相关文章

/