Connections between centrality and local monotonicity of certain functions on $C^*$-algebras

TL;DR

This paper characterizes central elements in $C^*$-algebras using local monotonicity of a broad class of functions, including exponential and power functions, unifying several previous results.

Contribution

It introduces a large class of functions that characterize centrality via local monotonicity in $C^*$-algebras, extending and unifying prior findings.

Findings

Characterization of central elements through local monotonicity.

Inclusion of exponential and power functions with exponent > 1.

Unification of previous results by Ogasawara, Pedersen, Wu, and Molnár.

Abstract

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 $a \leq b$ implies $f(a) \leq f(b).$ That is, we characterize centrality by local monotonicity of certain functions on $C^*$-algebras. Numerous former results (including works of Ogasawara, Pedersen, Wu, and Moln\'ar) are apparent consequences of our result.