Convex hulls of unitary orbits of normal elements in C⁎-algebras with tracial rank zero

Abstract

Let A be a unital separable simple C⁎-algebra with tracial rank zero and let x,y∈A be two normal elements. We show that x is in the closure of the convex hull of the unitary orbit of y if and only if there exists a sequence of unital completely positive linear maps φn from A to A such that the sequence φn(y) converges to x in norm and also approximately preserves the trace values. A purely measure theoretical description for normal elements in the closure of the convex hull of unitary orbit of y is also given. In the case that A has a unique tracial state some classical results about the closure of the convex hull of the unitary orbits in von Neumann algebras are proved to hold in the C⁎-algebraic setting.