Connections between centrality and local monotonicity of certain functions on C⁎-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⁎-algebra is central if and only if a≤b implies f(a)≤f(b). That is, we characterize centrality by local monotonicity of certain functions on C⁎-algebras. Numerous former results (including works of Ogasawara, Pedersen, Wu, and Molnár) are apparent consequences of our result.