Equality of ultraweak and operator-norm closures of convex hulls of unitary orbits

Determine whether, for every σ-finite von Neumann algebra M and every element y ∈ M (not necessarily self-adjoint), the ultraweak closure of the convex hull of the unitary orbit {uyu : u ∈ U(M)} equals its operator-norm closure; equivalently, ascertain whether cl_ultraweak(conv({uyu : u ∈ U(M)})) = cl_norm(conv({uyu : u ∈ U(M)})) holds for arbitrary y in σ-finite von Neumann algebras.

Background

The paper develops separation theorems that give metric characterizations of convex hull closures in von Neumann algebras and applies these to majorization theory. For self-adjoint elements, combining the authors’ metric characterization with the Hiai–Nakamura theorem shows that the ultraweak and operator-norm closures of convex hulls of unitary orbits coincide.

However, for general (non-self-adjoint) elements in σ-finite von Neumann algebras, the equivalence of these closures has not been established. The authors explicitly note that this question remains open, referencing related discussion in Hiai and Nakamura [6, p. 36].