Characterizations of centrality by local convexity of certain functions on $C^{\ast }$-algebras

We provide a function class which is useful to distinguish central and non-central elements of a $C^{\ast }$-algebra in the following sense: for each element $f$ of this function class, a self-adjoint element $a$ of a $C^{\ast }$-algebra is central if and only if $f$ is locally convex at $a\mathrm{.}$