Alberti–Uhlmann Problem on Hardy–Littlewood–Pólya Majorization

We fully describe the doubly stochastic orbit of a self-adjoint element in the noncommutative L₁-space affiliated with a semifinite von Neumann algebra, which answers a problem posed by Alberti and Uhlmann (Stochasticity and partial order: doubly stochastic maps and unitary mixing. VEB Deutscher Verlag der Wissenschaften, Berlin, 1982) in the 1980s, extending several results in the literature. It follows further from our methods that, for any σ-finite von Neumann algebra M equipped with a semifinite infinite faithful normal trace τ, there exists a self-adjoint operator y∈ L₁(M, τ) such that the doubly stochastic orbit of y does not coincide with the orbit of y in the sense of Hardy–Littlewood–Pólya, which confirms a conjecture by Hiai (J Math Anal Appl 127:18–48, 1987). However, we show that Hiai’s conjecture fails for non-σ-finite von Neumann algebras. The main result of the present paper also answers the (noncommutative) infinite counterparts of problems due to Luxemburg (Proc Symp Anal Queen’s univ 83–144, 1967) and Ryff (Pac J Math 13:1379–1386, 1963) in the 1960s.