Convex hulls of unitary orbits of normal elements in $C^{\ast }$-algebras with tracial rank zero

Abstract

Let $A$ be a unital separable simple $C^{\ast }$-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 ${\varphi }_n$ from $A$ to $A$ such that the sequence ${\varphi }_n(y)$ 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 $C^{\ast }$-algebras setting.

arXiv commentA reference is corrected