An order characterization of commutativity for $C^{\ast }$-algebras
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- Proc. Amer. Math. Soc. 129 (2001), 983-987 Request permission
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 $\mathcal {A}$.References
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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
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 983-987
- MSC (2000): Primary 46L05
- DOI: https://doi.org/10.1090/S0002-9939-00-05724-5
- MathSciNet review: 1814137