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

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 $\mathcal{A}$.