Connections between centrality and local monotonicity of certain functions on $C^{\ast }$-algebras

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 $C^{\ast }$-algebra is central if and only if $a\le b$ implies $f(a)\le f(b).$ That is, we characterize centrality by local monotonicity of certain functions on $C^{\ast }$-algebras. Numerous former results (including works of Ogasawara, Pedersen, Wu, and Molnár) are apparent consequences of our result.

v2: major revision, stronger result, title changed. v3: minor improvements. v4: published version

References in corpus (1)

• Characterization of central elements of operator algebras by inequalities