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arXiv · 2004.03197

Alberti--Uhlmann problem on Hardy--Littlewood--P\'{o}lya majorization

Jinghao Huang·Fedor Sukochev
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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--P\'{o}lya, 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.

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Jinghao Huang, Fedor Sukochev. 2020-04-07. Alberti--Uhlmann problem on Hardy--Littlewood--P\'{o}lya majorization. https://arxiv.org/abs/2004.03197

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