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Fedor Sukochev

Publications and source records attributed to Fedor Sukochev.

At least 37 records · Page 2Linked to original sources
arXiv2025

Noncommutative sharp dual Doob inequalities

Let $(x_k)_{k=1}^n$ be positive elements in the noncommutative Lebesgue space $L_p(\mathcal{M})$, and let $(\mathcal{E}_k)_{k=1}^n$ be a sequence of conditional expectations with respect to an increasing subalgebras $(\mathcal{M}_n)_{k\geq1}$ of the finite von Neumann algebra $\mathcal{M}$. We establish the following sharp noncommutative dual Doob inequalities: \begin{equation*} \Big\| \sum_{k=1}^nx_k\Big\|_{L_p(\mathcal{M})}\leq \frac{1}{p} \Big\| \sum_{k=1}^n\mathcal{E}_k(x_k)\Big\|_{L_p(\mathcal{M})},\quad 0<p\leq 1, \end{equation*} and \begin{equation*} \Big\| \sum_{k=1}^n\mathcal{E}_k(x_k)\Big\|_{L_p(\mathcal{M})}\leq p\Big\| \sum_{k=1}^nx_k\Big\|_{L_p(\mathcal{M})},\quad 1\leq p\leq 2. \end{equation*} As applications, we obtain several noncommutative martingale inequalities with better constants.

math.OA↗
arXiv2024

Kuroda's theorem for $n$-tuples in semifinite von Neumann algebras

Let $\mathcal{M}$ be a semifinite von Neumann algebra and let $E$ be a symmetric function space on $(0,\infty)$. Denote by $E(\mathcal{M})$ the non-commutative symmetric space of measurable operators affiliated with $\mathcal{M}$ and associated with $E.$ Suppose $n\in \mathbb{N}$ and $E\cap L_{\infty}\not\subset L_{n,1}$, where $L_{n,1}$ is the Lorentz function space with the fundamental function $φ(t)=t^{1/n}$. We prove that for every $\varepsilon>0$ and every commuting self-adjoint $n$-tuple $(α(j))_{j=1}^n,$ where $α(j)$ is affiliated with $\mathcal{M}$ for each $1\leq j\leq n,$ there exists a commuting $n$-tuple $(δ(j))_{j=1}^n$ of diagonal operators affiliated with $\mathcal{M}$ such that $\max\{\|α(j)-δ(j)\|_{E(\mathcal{M})},\|α(j)-δ(j)\|_{\infty}\}<\varepsilon$ for each $1\le j\le n$. In the special case when $\mathcal{M}=B(H)$, our results yield the classical Kuroda and Bercovici-Voiculescu theorems.

math.OA↗
arXiv2024

Spectral asymptotic formula of Bessel--Riesz commutator

Let $R_{λ,j}$ be the $j$-th Bessel--Riesz transform, where $n\geq 1$, $λ>0$, and $j=1,\ldots,n+1$. In this article, we establish a Weyl type asymptotic for $[M_f,R_{λ,j}]$, the commutator of $R_{λ,j}$ with multiplication operator $M_f$, based on building a preliminary result that the endpoint weak Schatten norm of $[M_f,R_{λ,j}]$ can be characterised via homogeneous Sobolev norm $\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})$ of the symbol $f$. Specifically, the asymptotic coefficient is equivalent to $\|f\|_{\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})}.$ Our main strategy is to relate Bessel--Riesz commutator to classical Riesz commutator via Schur multipliers, and then to establish the boundedness of Schur multipliers.

math.FA↗
arXiv2024

A $C^{*}$-Algebraic Approach To Principal Symbol Calculus On Filtered Manifolds

From the viewpoint of $*$-homomorphism on $C^{*}$-algebras, we establish the principal symbol mapping for filtered manifolds which are locally isomorphic to stratified Lie groups. Let $\mathbb{G}$ be a stratified Lie group, and let $M$ be a filtered manifold with a $\mathbb{G}$-atlas and a smooth positive density $ν$. For the $C^{*}$-algebra bundle $E_{hom}$ of $M$ constructed from quasi-Riesz transforms on $\mathbb{G}$, we show that there exists a surjective $*$-homomorphism $${\rm sym}_{M}:Π_{M}\to C_{b}(E_{hom})$$ such that $${\rm ker}({\rm sym}_{M})=\mathcal{K}(L_{2}(M,ν))\subset Π_{M}$$ where the domain $Π_{M}\subset\mathcal{B}(L_{2}(M,ν))$ is a $C^{*}$-algebra and $C_{b}(E_{hom})$ is the $C^{*}$-algebra of bounded continuous sections of $E_{hom}$. Especially, we do not make any assumptions on the lattice of the osculating group of $M$ or the assumption of compactness on manifolds in \cite{DAO3,DAO4}.

math.OA↗
arXiv2024

Gagliardo-Nirenberg interpolation inequality for symmetric spaces on Noncommutative torus

Let $E(\mathbb{T}^{d}_θ),F(\mathbb{T}^{d}_θ)$ be two symmetric operator spaces on noncommutative torus $\mathbb{T}^{d}_θ$ corresponding to symmetric function spaces $E,F$ on $(0,1)$. We obtain the Gagliardo--Nirenberg interpolation inequality with respect to $\mathbb{T}^{d}_θ$: if $G=E^{1-\frac{l}{k}}F^{\frac{l}{k}}$ with $ 0\leq l\leq k$ and if the Cesàro operator is bounded on $E$ and $F$, then \begin{align*} \|\nabla^lx\|_{G(\mathbb{T}^{d}_θ)}\leq 2^{3\cdot 2^{k-2}-2}(k+1)^d\|C\|_{E\to E}^{1-\frac{l}{k}}\|C\|_{F\to F}^{\frac{l}{k}}\|x\|_{E(\mathbb{T}^{d}_θ)}^{1-\frac{l}{k}}\|\nabla^kx\|_{F(\mathbb{T}^{d}_θ)}^{\frac{l}{k}},\; x\in W^{k,1}(\mathbb{T}^{d}_θ), \end{align*} where $W^{k,1}(\mathbb{T}^{d}_θ)$ is the Sobolev space on $\mathbb{T}^{d}_θ$ of order $k\in\mathbb{N}$. Our method is different from the previous settings, which is of interest in its own right.

math.FA↗
arXiv2024

Diagonality modulo symmetric spaces in semifinite von Neumann algebras

In the study on the diagonality of an $n$-tuple $α=(α(j))_{j=1}^n$ of commuting self-adjoint operators modulo a given $n$-tuple $Φ=(\mathcal{J}_1,\ldots,\mathcal{J}_n)$ of normed ideals in $B(H)$, Voiculescu introduced the notion of quasicentral modulus $k_Φ(α)$ and proved that $α$ is diagonal modulo $(\mathcal{J}_1,\ldots,\mathcal{J}_n)$ if and only if $k_Φ(α)=0.$ We prove that the same assertion holds true when $B(H)$ is replaced with a $σ$-finite semifinite von Neumann algebra $\mathcal{M}$, and $\mathcal{J}_1,\ldots,\mathcal{J}_n$ are replaced with symmetric spaces $E_1(\mathcal{M}),\ldots,E_n(\mathcal{M})$ associated with $\mathcal{M}.$

math.OA↗
arXiv2024

Endpoint Schatten class properties of commutators

We study trace ideal properties of the commutators $[(-Δ)^{\fracε{2}},M_f]$ of a power of the Laplacian with the multiplication operator by a function $f$ on $\mathbb R^d$. For a certain range of $ε\in\mathbb R$, we show that this commutator belongs to the weak Schatten class $\mathcal L_{\frac d{1-ε},\infty}$ if and only if the distributional gradient of $f$ belongs to $L_{\frac d{1-ε}}$. Moreover, in this case we determine the asymptotics of the singular values. Our proofs use, among other things, the tool of Double Operator Integrals.

math.FA↗
arXiv2023

Commutator estimates for normal operators in factors with applications to derivations

For a normal measurable operator $a$ affiliated with a von Neumann factor $\mathcal{M}$ we show: If $\mathcal{M}$ is infinite, then there is $λ_0\in \mathbb{C}$ so that for $\varepsilon>0$ there are $u_{\varepsilon}=u_{\varepsilon}^*$, $v_{\varepsilon}\in \mathcal{U}(\mathcal{M})$ with $$v_\varepsilon|[a,u_\varepsilon]|v_\varepsilon^*\geq(1-\varepsilon)(|a-λ_0\textbf{1}|+u_\varepsilon|a-λ_0\textbf{1}|u_\varepsilon).$$ If $\mathcal{M}$ is finite, then there is $λ_0\in\mathbb{C}$ and $u,v\in\mathcal{U}(\mathcal{M})$ so that $$v|[a,u]|v^*\geq \frac{\sqrt{3}}{2}(|a-λ_0\textbf{1}|+u|a-λ_0\textbf{1}|u^*).$$ These bounds are optimal for infinite factors, II$_1$-factors and some I$_n$-factors. Furthermore, for finite factors applying $\|\cdot\|_{1}$-norms to the inequality provides estimates on the norm of the inner derivation $δ_{a}:\mathcal{M}\to L_1(\mathcal{M},τ)$ associated to $a$. While by [3,Theorem 1.1] it is known for finite factors and self-adjoint $a\in L_1(\mathcal{M},τ)$ that $\|δ_{a}\|_{\mathcal{M}\to L_1(\mathcal{M},τ)} = 2\min_{z\in \mathbb{C}}\|a-z\|_{1}$, we present concrete examples of finite factors $\mathcal{M}$ and normal operators $a\in \mathcal{M}$ for which this fails.

math.OA↗
arXiv2023

Local invariants of conformally deformed non-commutative tori II: multiple operator integrals

We explicitly compute the local invariants (heat kernel coefficients) of a conformally deformed non-commutative $d$-torus using multiple operator integrals. We derive a recursive formula that easily produces an explicit expression for the local invariants of any order $k$ and in any dimension $d$. Our recursive formula can conveniently produce all formulas related to the modular operator, which before were obtained in incremental steps for $d\in\{2,4\}$ and $k\in\{0,2,4\}$. We exemplify this by writing down some known ($k=2$, $d=2$) and some novel ($k=2$, $d\geq 3$) formulas in the modular operator.

math.OA↗
arXiv2023

Innerness of derivations into noncommutative symmetric spaces is determined commutatively

Let $E=E(0,\infty)$ be a symmetric function space and $E(\mathcal{M},τ)$ be a symmetric operator space associated with a semifinite von Neumann algebra with a faithful normal semifinite trace. Our main result identifies the class of spaces $E$ for which every derivation $δ:\mathcal{A}\to E(\mathcal{M},τ)$ is necessarily inner for each $C^*$-subalgebra $\mathcal{A}$ in the class of all semifinite von Neumann algebras $\mathcal{M}$ as those with the Levi property.

math.OA↗
arXiv2023

Hermitian operators and isometries on symmetric operator spaces

Let $\mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal trace $τ$. Let $E(\mathcal{M},τ) $ be a symmetric operator space affiliated with $ \mathcal{M} $, whose norm is order continuous and is not proportional to the Hilbertian norm $\left\|\cdot\right\|_2$ on $L_2(\mathcal{M},τ)$. We obtain general description of all bounded hermitian operators on $E(\mathcal{M},τ)$. This is the first time that the description of hermitian operators on asymmetric operator space (even for a noncommutative $L_p$-space) is obtained in the setting of general (non-hyperfinite) von Neumann algebras. As an application, we resolve a long-standing open problem concerning the description of isometries raised in the 1980s, which generalizes and unifies numerous earlier results.

math.OA↗
arXiv2022

Asymptotics of singular values for quantum derivatives

We obtain Weyl type asymptotics for the quantised derivative $\dbar f$ of a function $f$ from the homgeneous Sobolev space $\dot{W}^1_d(\mathbb{R}^d)$ on $\mathbb{R}^d.$ The asymptotic coefficient $\|\nabla f\|_{L_d(\mathbb R^d)}$ is equivalent to the norm of $\dbar f$ in the principal ideal $\mathcal{L}_{d,\infty},$ thus, providing a non-asymptotic, uniform bound on the spectrum of $\dbar f.$ Our methods are based on the $C^{\ast}$-algebraic notion of the principal symbol mapping on $\mathbb{R}^d$, as developed recently by the last two authors and collaborators.

math.SP↗
arXiv2022

Isomorphic classification of $L_{p,q}$-spaces, II

This is a continuation of the papers [Kuryakov-Sukochev, JFA, 2015] and [Sadovskaya-Sukochev, PAMS, 2018], in which the isomorphic classification of $L_{p,q}$, for $1< p<\infty$, $1\le q<\infty$, $p\ne q $, on resonant measure spaces, has been obtained. The aim of this paper is to give a complete isomorphic classification of $L_{p,q}$-spaces on general $σ$-finite measure spaces. Towards this end, several new subspaces of $L_{p,q}(0,1)$ and $L_{p,q}(0,\infty)$ are identified and studied.

math.FA↗
arXiv2022

Optimality of Cwikel-Solomyak estimates in Orlicz spaces

We discuss the optimality of Cwikel-Solomyak estimates for the uniform operator norm and establish optimality of M.Z Solomyak's results \cite{Solomyak1995} within the class of Orlicz spaces. Our methods are based on finding the optimal version of the Sobolev embedding theorem.

math.FA↗
arXiv2022

Endpoint weak Schatten class estimates and trace formula for commutators of Riesz transforms with multipliers on Heisenberg groups

Along the line of singular value estimates for commutators by Rochberg-Semmes, Lord-McDonald-Sukochev-Zanin and Fan-Lacey-Li, we establish the endpoint weak Schatten class estimate for commutators of Riesz transforms with multiplication operator $M_f$ on Heisenberg groups via homogeneous Sobolev norm of the symbol $f$. The new tool we exploit is the construction of a singular trace formula on Heisenberg groups, which, together with the use of double operator integrals, allows us to bypass the use of Fourier analysis and provides a solid foundation to investigate the singular values estimates for similar commutators in general stratified Lie groups.

math.FA↗
arXiv2022

Operator $θ$-Hölder functions with respect to $\left\|\cdot\right\|_p$, $0< p\le \infty$

Let $θ\in(0,1)$ and $(\mathcal{M},τ)$ be a semifinite von Neumann algebra. We consider the function spaces introduced by Sobolev (denoted by $S_{d,θ}$), showing that there exists a constant $d>0 $ depending on $p$, $0 1$, we obtain a reverse of the Birman-Koplienko-Solomjak inequality, which extends a couple of existing results on fractional powers $t\mapsto t^θ$ by Ando et al.

math.FA↗
arXiv2022

The Witten index and the spectral shift function

In \cite{APSIII} Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin-Salamon \cite{RS95}. In \cite{GLMST}, a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper \cite{Pu08}. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and {its perturbation by a relatively trace-class operator}. In this paper we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher $p^{th}$ Schatten class condition for $0\leq p<\infty$, thus allowing differential operators on manifolds of any dimension $d<p+1$. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by \cite{BCPRSW, CGK16}. We illustrate our results using Dirac type operators on $L^2(\bbR^d)$ for arbitrary $d\in\bbN$. In this setting our main result substantially extends \cite[Theorem 3.5]{CGGLPSZ16}, where the case $d=1$ was treated.

math.FA↗