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
蒋闰良
. C^*-代数交换性简谈(英)[J]. 华东师范大学学报（自然科学版）, 2016
, 2016(2)
: 30
-34
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DOI: 2016.02.004
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.
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