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An order characterization of commutativity for -algebras

Authors:	Wei Wu

Institution:	Institute of Mathematics, Academia Sinica, Beijing 100080, China

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 in $\mathcal{A}$ .

Keywords:	Commutativity for $C^{\ast }$-algebras operator-monotonic increasing function positive element

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