Closed convex hulls of unitary orbits in von Neumann algebras

Abstract

Let M \mathcal {M} be a von Neumann algebra. The distance dist ⁡ ( x , co ⁡ U ( y ) ) \operatorname {dist} (x,\operatorname {co} \mathcal {U}(y)) between x x and co ⁡ U ( y ) \operatorname {co} \mathcal {U}(y) for selfadjoint operators x x , y ∈ M y \in \mathcal {M} and the distance dist ⁡ ( φ , co ⁡ U ( ψ ) ) \operatorname {dist} (\varphi ,\operatorname {co} \mathcal {U}(\psi )) between φ \varphi and co ⁡ U ( ψ ) \operatorname {co} \mathcal {U}(\psi ) for selfadjoint elements φ \varphi , ψ ∈ M ∗ \psi \in {\mathcal {M}_*} are exactly estimated, where co ⁡ U ( y ) \operatorname {co} \mathcal {U}(y) and co ⁡ U ( ψ ) \operatorname {co} \mathcal {U}(\psi ) are the convex hulls of the unitary orbits of y y and ψ \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 M \mathcal {M} is a von Neumann algebra on a separable Hilbert space, the above cases altogether are combined under the direct integral decomposition of M \mathcal {M} into factors. As a result, it is known that if M \mathcal {M} is σ \sigma -finite and x ∈ M x \in \mathcal {M} is selfadjoint, then co ¯ U ( x ) = co ¯ w U ( x ) \overline {\operatorname {co} } \mathcal {U}(x) = {\overline {\operatorname {co} } ^{\mathbf {w}}}\mathcal {U}(x) where co ¯ U ( x ) \overline {\operatorname {co} } \mathcal {U}(x) and co ¯ w U ( x ) {\overline {\operatorname {co} } ^{\mathbf {w}}}\mathcal {U}(x) are the closures of co ⁡ U ( x ) \operatorname {co} \mathcal {U}(x) in norm and in the weak operator topology, respectively.