Convex hulls of unitary orbits of normal elements in $C^*$-algebras with tracial rank zero

TL;DR

This paper characterizes when a normal element in a unital simple $C^*$-algebra with tracial rank zero lies in the convex hull of a unitary orbit, using completely positive maps and trace preservation.

Contribution

It provides a new characterization of convex hulls of unitary orbits in $C^*$-algebras with tracial rank zero, extending classical results to this setting.

Findings

Characterization of elements in the convex hull via completely positive maps

Measure-theoretic description of normal elements in the convex hull

Extension of von Neumann algebra results to $C^*$-algebras with unique trace

Abstract

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.