Characterizations of centrality by local convexity of certain functions on $C^*$-algebras

TL;DR

This paper introduces a specific class of functions that can identify central elements in a $C^*$-algebra by examining local convexity properties at those elements, providing a new characterization tool.

Contribution

It presents a novel function class that characterizes centrality in $C^*$-algebras through local convexity conditions.

Findings

A function class distinguishes central from non-central elements.

Centrality is characterized by local convexity of functions at elements.

The approach offers a new perspective on the structure of $C^*$-algebras.

Abstract

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