Alberti--Uhlmann problem on Hardy--Littlewood--Pólya majorization

Abstract

We fully describe the doubly stochastic orbit of a self-adjoint element in the noncommutative $L_1$-space affiliated with a semifinite von Neumann algebra, which answers a problem posed by Alberti and Uhlmann in the 1980s, extending several results in the literature. It follows further from our methods that, for any $\sigma$-finite von Neumann algebra $\mathcal{M}$ equipped a semifinite infinite faithful normal trace $\tau$, there exists a self-adjoint operator $y\in L_1(\mathcal{M},\tau )$ 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. However, we show that Hiai's conjecture fails for non-$\sigma$-finite von Neumann algebras. The main result of the present paper also answers the (noncommutative) infinite counterparts of problems due to Luxemburg and Ryff in the 1960s.

References in corpus (2)

• Thompson's theorem for compact operators and diagonals of unitary operators
• Closed Convex Hulls of Unitary Orbits in C${}^{\ast }$-Algebras of Real Rank Zero