Connections between centrality and local monotonicity of certain functions on $C^{\ast }$-algebras

We introduce a quite large class of functions (including the exponential function and the power functions with exponent greater than one), and show that for any element $f$ of this function class, a self-adjoint element $a$ of a $C^{\ast }$-algebra is central if and only if $a\le b$ implies $f(a)\le f(b)\mathrm{.}$ That is, we characterize centrality by local monotonicity of certain functions on $C^{\ast }$-algebras. Numerous former results (including works of Ogasawara, Pedersen, Wu, and Moln'ar) are apparent consequences of our result.