华东师范大学学报(自然科学版) ›› 2016, Vol. 2016 ›› Issue (2): 30-34.doi: 2016.02.004

• 数学 • 上一篇    下一篇

C^*-代数交换性简谈(英)

交换$C^*$-代数有许多特征. 在本文中,证明了~$C^*$-代数~$\mathcal{A}$~是非交换的当且仅当其包络 冯诺依曼代数~$\mathcal{A}''$~中有一个~$C^*$-子代数~$\mathcal{B}$, $\mathcal{B}$~$*$-同构于2阶矩阵代数~$\mathrm M_2(\C)$. 基于这个性质,又可以得到一些旧命题的新证明方法.   

  1. 上海师范大学~~数学系, 上海 200234
  • 收稿日期:2015-03-31 出版日期:2016-03-25 发布日期:2016-07-25
  • 作者简介:蒋闰良, 男, 博士后, 研究方向为算子代数. E-mail:Eugene_Jiang@126.com

A note on the commutativity of {C}^*-algebras

There are many characterizations for commutative C^*-algebras. In this note, we prove that a C^*-algebra \mathcal{A} is not commutative if and only if there is a C^*-subalgebra \mathcal{B} in \mathcal{A}'' (the enveloping Von Neumann algebra of mathcal{A}) such that mathcal{B} is -isomorphic to \mathrm M_2(\mathcal{\textbf{C}}). In terms of this result, we can recover some characterizations for the commutativity of C^*-algebras appeared before   

  • Received:2015-03-31 Online:2016-03-25 Published:2016-07-25
  • Contact: 蒋闰良, 男, 博士后, 研究方向为算子代数. E-mail: Eugene_Jiang@126.com

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摘要： 交换C^*-代数有许多特征. 在本文中,证明了~C^*-代数~\mathcal{A}~是非交换的当且仅当其包络 冯诺依曼代数~\mathcal{A}''~中有一个~C^*-子代数~\mathcal{B}, \mathcal{B}-同构于2阶矩阵代数~\mathrm M_2(\C). 基于这个性质,又可以得到一些旧命题的新证明方法

关键词: 交换~C^*-代数, 包络冯诺依曼代数

Abstract: There are many characterizations for commutative C^*-algebras. In this note, we prove that a C^*-algebra $\mathcal{A} is not commutative if and only if there is a C^*-subalgebra \mathcal{B} in \mathcal{A}'' (the enveloping Von Neumann algebra of mathcal{A}) such that mathcal{B} is-isomorphic to mathrm M_2(\mathcal{\textbf{C}}). In terms of this result, we can recover some characterizations for the commutativity of C^-algebras appeared before.

Key words: commutative C^*-algebras, enveloping Von Neumannalgebra

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