Closed convex hulls of unitary orbits in C⁎-algebras of real rank zero

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