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 full of the unitary obit of y if and only if there exists a sequence of unital completely positive linear maps ϕ n from A to A such that the sequence ϕ n (y) convergent to x in norm and also approximately preserves the trace values. A purely measure theoretical description for normal elements in the closure of 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 be hold in C * -algebras setting.