von Neumann代数中的*-偏序

张欣培, 史维娟, 吉国兴

数学学报 ›› 2017, Vol. 60 ›› Issue (1) : 19-30.

PDF(485 KB)
PDF(485 KB)
数学学报 ›› 2017, Vol. 60 ›› Issue (1) : 19-30. DOI: 10.12386/A2017sxxb0002
论文

von Neumann代数中的*-偏序

    张欣培, 史维娟, 吉国兴
作者信息 +

Star Partial Order in a von Neumann Algebra

    Xin Pei ZHANG, Wei Juan SHI, Guo Xing JI
Author information +
文章历史 +

摘要

H是复Hilbert空间,BH)是H上的有界线性算子全体组成的代数,MBH)是von Neumann代数,“≤”表示M中的*-偏序,即A,BM,若A*A=A*B,AA*=BA*,则AB.本文研究了von Neumann代数中*-偏序的上确界和下确界,证明了von Neumann代数M的子集关于*-偏序的上、下确界和BH)中的上、下确界一致.同时,给出了M的*-偏序遗传子空间的表示,证明了弱*闭子空间AM,满足AM,BA,由AB可得AA,当且仅当存在唯一具有相同中心投影的投影对E,FM,使得A=EMF.

Abstract

Let H be a complex Hilbert space and B(H) the algebra of all bounded linear operators on H.Let MB(H) be a von Neumann algebra and " the star partial order in M, that is, for A, BM, then we say that AB if A*A=A*B and AA*=BA*.It is proved that the supremum and infimum of a subset in M with respect to the star partial order are the same as in B(H).Moreover, we give the representation of a star partial order-hereditary subspace in M, that is, a W* closed nonzero subspace A in M is star partial order-hereditary, which means that for any AM and BA, AA whenever AB, if and only if there is a unique pair of nonzero projections E and F which have the same central carrier in M such that A=EMF.

关键词

von Neumann代数 / *-偏序 / 上确界 / 下确界 / *-偏序遗传子空间

Key words

von Neumann algebra / star partial order / supremum / infimum / star partial order-hereditary subspace

引用本文

导出引用
张欣培, 史维娟, 吉国兴. von Neumann代数中的*-偏序. 数学学报, 2017, 60(1): 19-30 https://doi.org/10.12386/A2017sxxb0002
Xin Pei ZHANG, Wei Juan SHI, Guo Xing JI. Star Partial Order in a von Neumann Algebra. Acta Mathematica Sinica, Chinese Series, 2017, 60(1): 19-30 https://doi.org/10.12386/A2017sxxb0002

参考文献

[1] Antezana J., Cano C., A note on the star order in Hilbert spaces, Linear and Multilinear Algebra, 2010, 58(8):1037-1051.
[2] Baksalary J. K., Mitra S. K., Left-star and right-star partial orderings, Linear Algebra and Its Applications, 1991, 149(1):73-89.
[3] Baksalary J. K., Further properties of the star, left-star, right-star and minus partial orderings, Linear Algebra and Its Applications, 2003, 375(1):83-94.
[4] Baksalary J. K., Relationships between partial orders of matrices and their powers, Linear Algebra and Its Applications, 2004, 379(1):277-287.
[5] Bohata M., Hamhalter J., Nonlinear maps on von Neumann algebras preserving the star order, Linear and Multilinear Algebra, 2013, 61(61):998-1009
[6] Deng C., Some properties on the star order of bounded operators, Journal of Mathematical Analysis and Applications, 2014, 423(1):32-40.
[7] Deng C., Yu A., Some relations of projection and star order in Hilbert space, Linear Algebra and Its Applications, 2015, 474(1):158-168.
[8] Deng C., Du H., Common complements of two subspaces and an answer to Groβ's question, Acta Mathematica Sinica, in Chinese, 2006, 49(5):1099-1112.
[9] Dolinar G., Guterman A., Marovt J., Automorphisms of K(H) with respect to the star partial order, Operators and Matrices, 2013, 1(1):225-239.
[10] Dolinar G., Marovt J., Star partial order on B(H), Linear Algebra and Its Applications, 2011, 434:319-321.
[11] Dragana S., Cvetkovic-Ilic, Partial orders on B(H), Linear Algebra and Its Applications, 2015, 481:115-130.
[12] Hartwig R. E., Drazin M. P., Lattice properties of the star order for complex matrices, American Journal of Mathematics, 1982, 86(2):359-378.
[13] He H., Ji G., Minus partial order-hereditary subspaces in B(H)(in Chinese), Advances in Mathematics, 2013, 42(5):701-705.
[14] Jose S., Sivakumar K. C., On partial order of Hilbert space operators, Linear and Multilinear Algebra, 2015, 63(7):1423-1441.
[15] Kadison R. V., Ringrose J. R., Fundamentals of the Theory of Operator Algebras Vol. I, Academic Press Inc.(London) LTD., 1983.
[16] Marko S., Properties of the star supremum for arbitrary Hilbert space operators, Journal of Mathematical Analysis and Applications, 2016, 441(1):446-461.
[17] Pang Y., Du H., Characters and properties of partial orders of operators, in Chinese, Journal of Mathematical Research and Exposition, 2007, 27(4):889-895.
[18] Šemrl P., Automorphisms of B(H) with respect to minus partial order, Journal of Mathematical Analysis and Applications, 2010, 369:205-213.
[19] Xi C., Ji G., Additive maps preserving the star partial order on B(H), Communications in Mathematical Research, 2015, 31(1):89-96.
[20] Xu X., Du H., Fang X., Li Y., The supremum of linear operators for the *-order, Linear Algebra and Its Applications, 2010, 433(11):2198-2207.
[21] Zhang Q., Ji G., Star partial order-hereditary subspaces in B(H), Operators and Matrices, 2014, 8(3):683-690.

基金

国家自然科学基金资助项目(11371233)

PDF(485 KB)

Accesses

Citation

Detail

段落导航
相关文章

/