arXiv subjects

Jinghao Huang

Publications and source records attributed to Jinghao Huang.

39 records · Page 3Linked to original sources

arXiv2020

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

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 $σ$-finite von Neumann algebra $\mathcal{M}$ equipped a semifinite infinite faithful normal trace $τ$, there exists a self-adjoint operator $y\in L_1(\mathcal{M},τ)$ 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. 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 and Ryff in the 1960s.

math.FA↗

arXiv2020

Non-existence of translation-invariant derivations on algebras of measurable functions

Let $S(0,1)$ be the $*$-algebra of all classes of Lebesgue measurable functions on the unit interval $(0,1)$ and let $(\mathcal{A},\left\|\cdot \right\|_\mathcal{A})$ be a complete symmetric $Δ$-normed $*$-subalgebra of $S(0,1)$, in which simple functions are dense, e.g., $L_\infty (0,1)$, $L_{\log}(0,1)$, $S(0,1)$ and the Arens algebra $L^ω(0,1)$ equipped with their natural $Δ$-norms. We show that there exists no non-trivial derivation $ δ: \mathcal{A} \to S(0,1)$ commuting with all dyadic translations of the unit interval. Let $\mathcal{M}$ be a type $II$ (or $I_\infty$) von Neumann algebra, $\mathcal{A}$ be its abelian von Neumann subalgebra, let $S(\mathcal{M})$ be the algebra of all measurable operators affiliated with $\mathcal{M}$. We show that any non-trivial derivation $δ:\mathcal{A} \to S(\mathcal{A})$ can not be extended to a derivation on $S(\mathcal{M})$. In particular, we answer an untreated question in \cite{BKS1}.

math.FA↗

arXiv2019

Logarithmic submajorisation and order-preserving linear isometries

Let $\mathcal{E}$ and $\mathcal{F}$ be symmetrically $Δ$-normed (in particular, quasi-normed) operator spaces affiliated with semifinite von Neumann algebras $\mathcal{M}_1$ and $\mathcal{M}_2$, respectively. We establish a noncommutative version of Abramovich's theorem \cite{A1983}, which provides the general form of normal order-preserving linear operators $T:\mathcal{E} \stackrel{into}{\longrightarrow} \mathcal{F}$ having the disjointness preserving property. As an application, we obtain a noncommutative Huijsmans-Wickstead theorem \cite{Huijsmans_W}. By establishing the disjointness preserving property for an order-preserving isometry $T:\mathcal{E} \stackrel{into}{\longrightarrow} \mathcal{F}$, we obtain the existence of a Jordan $*$-monomorphism from $\mathcal{M}_1$ into $\mathcal{M}_2$ and the general form of this isometry, which extends and complements a number of existing results. In particular, we fully resolve the case when $\mathcal{F}$ is the predual of $\mathcal{M}_2$ and other untreated cases in [Sukochev-Veksler, IEOT, 2018].

math.FA↗