Closed Convex Hulls of Unitary Orbits in C${}^{\ast }$-Algebras of Real Rank Zero

Abstract

In this paper, we study closed convex hulls of unitary orbits in various C${}^{\ast }$-algebras. For unital C${}^{\ast }$-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${}^{\ast }$-algebras. Combining these ideas with the Dixmier property, we demonstrate unital, infinite dimensional C${}^{\ast }$-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${}^{\ast }$-algebras.

References in corpus (4)

• Diagonals of self-adjoint operators
• Crossed products of the Cantor set by free minimal actions of Z^d
• Numerical Ranges in II${}_1$ Factors
• Unitary orbits of self-adjoint operators in simple Z-stable C*-algebras

Cited by in corpus (3)

• The Dixmier property and tracial states for C*-algebras
• Closed convex hulls of unitary orbits in certain simple real rank zero C${}^{\ast }$-algebras
• Alberti--Uhlmann problem on Hardy--Littlewood--Pólya majorization