Characterizations of centrality in $C^\ast$-algebras via local convexity of functions

In this paper, we give a characterization of central elements in a $C^*$-algebra $\mathcal{A}$ in terms of a local property of maps on $\mathcal{A}$ given by the functional calculus. We prove that if $f$ denotes one of the functions $x\mapsto\exp(x),\ x\mapsto x^3\ (x\in\mathbb{R})$, a self-adjoint element $a\in\mathcal{A}$, which is also positive in the case where $f$ is the latter map, is central if and only if $f$ is locally convex at $a$.