Characterizations of Centrality in {varvec{C}}^*-algebras via Local Monotonicity and Local Additivity of Functions

Gergő Nagy

doi:10.1007/s00020-019-2526-2

Abstract

In this paper, we give two characterizations of central elements in a C^*-algebra mathcal {A} in terms of local properties of maps on mathcal {A} given by the function calculus. We prove that for a strictly convex increasing function f defined on an open interval which is unbounded from above, an element ain mathcal {A} is central if and only if f is locally monotone at a. That result significantly improves similar theorems by Ogasawara, Pedersen, Wu, Molnár and Virosztek. An analogous statement on local additivity is also presented.

Highlights

Introduction and Statement of the MainResultsExtension of functions has a considerable literature and it gives ways to plug elements in them which originally did not belong to their domains
We give two characterizations of central elements in a C∗-algebra A in terms of local properties of maps on A given by the function calculus
We prove that for a strictly convex increasing function f defined on an open interval which is unbounded from above, an element a ∈ A is central if and only if f is locally monotone at a

Summary

Introduction and Statement of the Main Results

Extension of functions has a considerable literature and it gives ways to plug elements in them which originally did not belong to their domains. The continuous function calculus can be used to plug normal elements of a C∗-algebra in complex-valued continuous functions defined on the spectra of such elements It is a natural question whether certain properties of such functions are preserved, meaning that they still possess them on subsets of those operators. It turned out that for a large class of monotone functions, quite surprisingly, the preservation of this property can occur only when the underlying algebra is commutative. This shows that the monotonicity of certain maps on C∗-algebras defined by the continuous function calculus gives us important information about the algebraic structure of the algebra considered

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Proofs

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