Title：Doubly stochastic operators and Hardy-Littlewood-Pólya majorization
Speaker：Jinghao Huang（The University of New South Wales）
Time：16:00-17:00,October15
Location： Tecent Meeting, Tecent Meeting ID: 614 363 719
Abstract：It is well-known that the doubly stochastic orbit of a self-adjoint n×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. In particular, we confirm Hiai's conjecture, and completely resolve a problem raised by Luxemburg in 1967 on the description of extreme points of the Hardy-Littlewood-Polya orbit of an integrable function on a finite measure space.
