Journal of East China Normal University(Natural Sc ›› 2016, Vol. 2016 ›› Issue (2): 30-34.doi: 2016.02.004

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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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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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