Majorization in C*-algebras

TL;DR

This paper characterizes the convex hulls of unitary orbits of selfadjoint elements in unital C*-algebras using majorization and trace conditions, providing bounds on unitary conjugates needed for approximation in certain classes of algebras.

Contribution

It introduces a majorization framework for unbounded traces to characterize convex hulls and establishes bounds on unitary conjugates in specific classes of C*-algebras.

Findings

Characterization of convex hulls via majorization and traces.

Existence of bounds on unitary conjugates for certain C*-algebras.

Failure of these bounds in some 'badly behaved' C*-algebras.

Abstract

We investigate the closed convex hull of unitary orbits of selfadjoint elements in arbitrary unital C*-algebras. Using a notion of majorization against unbounded traces, a characterization of these closed convex hulls is obtained. Furthermore, for C*-algebras satisfying Blackadar's strict comparison of positive elements by traces or for collections of C*-algebras with a uniform bound on their nuclear dimension, an upper bound for the number of unitary conjugates in a convex combination required to approximate an element in the closed convex hull within a given error is shown to exist. This property, however, fails for certain "badly behaved" simple nuclear C*-algebras.