Local variants of the Dixmier property and weak centrality for C*-algebras

TL;DR

This paper explores localized variants of the Dixmier property and weak centrality in unital C*-algebras, providing spectral conditions and element-wise characterizations that extend classical notions.

Contribution

It introduces element-wise versions of the Dixmier property and weak centrality, along with spectral criteria for their characterization.

Findings

Characterization of Dixmier elements via spectral conditions

Description of elements approximable by central elements

Extension of weak centrality to element-wise variants

Abstract

We study variants of the Dixmier property that apply to elements of a unital C*-algebra, rather than to the C*-algebra itself. By a Dixmier element in a C*-algebra we understand one that can be averaged into a central element by means of a sequence of unitary mixing operators. Examples include all self-commutators and all quasinilpotent elements. We do a parallel study of an element-wise version of weak centrality, where the averaging to the centre is done using unital completely positive elementary operators (as in Magajna's characterization of weak centrality). We also obtain complete descriptions of more tractable sets of elements, where the corresponding averaging can be done arbitrarily close to the centre. This is achieved through several "spectral conditions", involving numerical ranges and tracial states.