Closed Convex Hulls of Unitary Orbits in C$^*$-Algebras of Real Rank Zero

TL;DR

This paper characterizes the closed convex hulls of unitary orbits in certain C$^*$-algebras, revealing structural properties and introducing majorization concepts for self-adjoint operators.

Contribution

It introduces a notion of majorization for self-adjoint operators in real rank zero C$^*$-algebras and describes convex hulls of unitary orbits in simple purely infinite cases.

Findings

Majorization describes convex hulls in real rank zero algebras.

Unital, simple, purely infinite C$^*$-algebras have convex hulls fully described.

Certain properties imply simplicity and uniqueness of the tracial state.

Abstract

In this paper, we study closed convex hulls of unitary orbits in various C$^*$-algebras. For unital C$^*$-algebras with real rank zero and a faithful tracial state determining equivalence of projections, a notion of majorization describes the closed convex hulls of unitary orbits for self-adjoint operators. Other notions of majorization are examined in these C$^*$-algebras. Combining these ideas with the Dixmier property, we demonstrate unital, infinite dimensional C$^*$-algebras of real rank zero and strict comparison of projections with respect to a faithful tracial state must be simple and have a unique tracial state. Also, closed convex hulls of unitary orbits of self-adjoint operators are fully described in unital, simple, purely infinite C$^*$-algebras.