Closed Convex Hulls of Unitary Orbits in von Neumann Algebras

Abstract

Let $\mathcal {M}$ be a von Neumann algebra. The distance $\operatorname {dist} (x,\operatorname {co} \mathcal {U}(y))$ between $x$ and $\operatorname {co} \mathcal {U}(y)$ for selfadjoint operators $x$, $y \in \mathcal {M}$ and the distance $\operatorname {dist} (\varphi ,\operatorname {co} \mathcal {U}(\psi ))$ between $\varphi$ and $\operatorname {co} \mathcal {U}(\psi )$ for selfadjoint elements $\varphi$, $\psi \in {\mathcal {M}_*}$ are exactly estimated, where $\operatorname {co} \mathcal {U}(y)$ and $\operatorname {co} \mathcal {U}(\psi )$ are the convex hulls of the unitary orbits of $y$ and $\psi$, respectively. This is done separately in the finite factor case, in the infinite semifinite factor case, and in the type III factor case. Simple formulas of distances between two convex hulls of unitary orbits are also given. When $\mathcal {M}$ is a von Neumann algebra on a separable Hilbert space, the above cases altogether are combined under the direct integral decomposition of $\mathcal {M}$ into factors. As a result, it is known that if $\mathcal {M}$ is $\sigma$-finite and $x \in \mathcal {M}$ is selfadjoint, then $\overline {\operatorname {co} } \mathcal {U}(x) = {\overline {\operatorname {co} } ^{\mathbf {w}}}\mathcal {U}(x)$ where $\overline {\operatorname {co} } \mathcal {U}(x)$ and ${\overline {\operatorname {co} } ^{\mathbf {w}}}\mathcal {U}(x)$ are the closures of $\operatorname {co} \mathcal {U}(x)$ in norm and in the weak operator topology, respectively.