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An order characterization of commutativity for $C^{\ast}$-algebras


Author: Wei Wu
Journal: Proc. Amer. Math. Soc. 129 (2001), 983-987
MSC (2000): Primary 46L05
DOI: https://doi.org/10.1090/S0002-9939-00-05724-5
Published electronically: October 10, 2000
MathSciNet review: 1814137
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Abstract: In this paper, we investigate the problem of when a $C^{\ast }$-algebra is commutative through operator-monotonic increasing functions. The principal result is that the function $e^{t}, t\in [0, \infty ),$ is operator-monotonic increasing on a $C^{\ast }$-algebra $\mathcal{A}$ if and only if $\mathcal{A}$ is commutative. Therefore, $C^{\ast }$-algebra $\mathcal{A}$ is commutative if and only if $e^{x+y}=e^{x}e^{y}$ in $\mathcal{A} \dot + \mathbf{C}$ for all positive elements $x, y$ in $\mathcal{A}$.


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Additional Information

Wei Wu
Affiliation: Institute of Mathematics, Academia Sinica, Beijing 100080, China
Address at time of publication: Department of Mathematics, East China Normal University, Shanghai 200062, China
Email: wwu@math03.math.ac.cn, wwu@math.ecnu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-00-05724-5
Keywords: Commutativity for $C^{\ast }$-algebras, operator-monotonic increasing function, positive element
Received by editor(s): November 4, 1998
Received by editor(s) in revised form: June 4, 1999
Published electronically: October 10, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society