C^*-代数交换性简谈(英)

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
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.