  Let $A$ be a unital separable simple $C^*$-algebra with tracial rank zero and
let $x, \, y\in A$ 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 $\phi_n$ from $A$ to $A$
such that the sequence $\phi_n(y)$ 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.
