On Alberti-Uhlmann Problem

It is well-known that the doubly stochastic orbit of a self-adjoint $n \times n$ matrix $A$ coincides with the Hardy-Littlewood-Polya orbit of $A$. Moreover, the set of all extreme points of the doubly stochastic orbit of $A$ coincides with the set of all matrices which are unitarily equivalent to $A$.
In 1982, Alberti and Uhlmann asked how to formulate a variant of these results in the setting of von Neumann algebras. In 1987, Hiai obtained that the doubly stochastic orbit coincides with the Hardy-Littlewood-Polya orbit in the setting of a finite von Neumann algebra equipped with a finite faithful normal trace, and he conjectured that the assumption that the trace is finite is sharp.
In this talk, we present an answer to the problem by Alberti and Uhlmann in the setting of general semifinite von Neumann algebras. We shall also discuss a Hiai’s conjecture, prompted by Alberti and Uhlmann’s problem and show when this conjecture holds and when it fails. The results are published in a joint paper (Huang, S.) in Communications in Mathematical Physics 2021.