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Quanhua Xu

Publications and source records attributed to Quanhua Xu.

At least 19 recordsLinked to original sources
arXiv2026

Completely Bounded Qusi-Norms, Their Mutiplicativity, and New Additivity Results of Quantum Channels

We obtain two new additivity results of quantum channels. The first one is the additivity of the channel Rényi information associated with the sandwiched Rényi divergence of order $α\in[\frac{1}{2},1)$. To prove this, we introduce the completely bounded $1\toα$ quasi-norms for completely positive maps, with $α\in[\frac{1}{2},1)$, and show that it is multiplicative. The additivity/multiplicativity derived here extends and complements the results of Devetak {\it et al} (Commun Math Phys 266:37-63, 2006) and Gupta and Wilde (Commun Math Phys 334:867-887, 2015), which deal with the case $α>1$. The second one is the additivity of the channel dispersion, which is a quantity related to the second-order behavior of quantum information tasks.

quant-ph↗
arXiv2025

An extension of Haagerup's reduction theorem with applications to subdiagonal subalgebras of general von Neumann algebras

We revisit Haagerup's enigmatic reduction theorem \cite[Theorems 2.1 \& 3.1]{HJX} showing how that theorem may be extended to general von Neumann algebras $\M$ equipped with an arbitrary faithful normal semifinite weight in a manner which faithfully captures the essence of the original. In contrast to the proposal in \cite[Remark 2.8]{HJX}, we show how in the non-$σ$-finite case the enlargement $\R=\M\rtimes\mathbb{Q}_D$ of $\M$ may be approximated by an increasing \emph{sequence} of \emph{expected} semifinite subalgebras. Using this revised version of the reduction theorem we may then all the applications of this theorem to $H^p$-spaces from the $σ$-finite case to general von Neumann algebras. Inspired by the theory of topologically ordered groups we then propose the even more general concept of approximately subdiagonal subalgebras which proves to be general enough to contain all group theoretic examples. This then forms the context for much of the study of Fredholm Toeplitz operators in the closing sections.

math.OA↗
arXiv2024

Holomorphic functional calculus and vector-valued Littlewood-Paley-Stein theory for semigroups

We study vector-valued Littlewood-Paley-Stein theory for semigroups of regular contractions $\{T_t\}_{t>0}$ on $L_p(Ω)$ for a fixed $1 0}$ is the Poisson semigroup subordinated to $\{T_t\}_{t>0}$. Let $\mathsf{L}^P_{c, q, p}(X)$ be the least constant $C$, and let $\mathsf{M}_{c, q}(X)$ be the martingale cotype $q$ constant of $X$. We show $$\mathsf{L}^{P}_{c,q, p}(X)\lesssim \max\big(p^{\frac1{q}},\, p'\big) \mathsf{M}_{c,q}(X).$$ Moreover, the order $\max\big(p^{\frac1{q}},\, p'\big)$ is optimal as $p\to1$ and $p\to\infty$. If $X$ is of martingale type $q$, the reverse inequality holds. If additionally $\{T_t\}_{t>0}$ is analytic on $L_p(Ω; X)$, the semigroup $\{P_t\}_{t>0}$ in these results can be replaced by $\{T_t\}_{t>0}$ itself. Our new approach is built on holomorphic functional calculus. Compared with all the previous, the new one is more powerful in several aspects: a) it permits us to go much further beyond the setting of symmetric submarkovian semigroups; b) it yields the optimal orders of growth on $p$ for most of the relevant constants; c) it gives new insights into the scalar case for which our orders of the best constants in the classical Littlewood-Paley-Stein inequalities for symmetric submarkovian semigroups are better than the previous by Stein. In particular, we resolve a problem of Naor and Young on the optimal order of the best constant in the above inequality when $X$ is of martingale cotype $q$ and $\{P_t\}_{t>0}$ is the classical Poisson and heat semigroups on $\mathbb{R}^d$.

math.FA↗
arXiv2022

Optimal orders of the best constants in the Littlewood-Paley inequalities

Let $\{\mathbb{P}_t\}_{t>0}$ be the classical Poisson semigroup on $\mathbb{R}^d$ and $G^{\mathbb{P}}$ the associated Littlewood-Paley $g$-function operator: $$G^{\mathbb{P}}(f)=\Big(\int_0^\infty t|\frac{\partial}{\partial t} \mathbb{P}_t(f)|^2dt\Big)^{\frac12}.$$ The classical Littlewood-Paley $g$-function inequality asserts that for any $1<p<\infty$ there exist two positive constants $\mathsf{L}^{\mathbb{P}}_{t, p}$ and $\mathsf{L}^{\mathbb{P}}_{c, p}$ such that $$ \big(\mathsf{L}^{\mathbb{P}}_{t, p}\big)^{-1}\big\|f\big\|_{p}\le \big\|G^{\mathbb{P}}(f)\big\|_{p} \le \mathsf{L}^{\mathbb{P}}_{c,p}\big\|f\big\|_{p}\,,\quad f\in L_p(\mathbb{R}^d). $$ We determine the optimal orders of magnitude on $p$ of these constants as $p\to1$ and $p\to\infty$. We also consider similar problems for more general test functions in place of the Poisson kernel. The corresponding problem on the Littlewood-Paley dyadic square function inequality is investigated too. Let $Δ$ be the partition of $\mathbb{R}^d$ into dyadic rectangles and $S_R$ the partial sum operator associated to $R$. The dyadic Littlewood-Paley square function of $f$ is $$S^Δ(f)=\Big(\sum_{R\inΔ} |S_R(f)|^2\Big)^{\frac12}.$$ For $1<p<\infty$ there exist two positive constants $\mathsf{L}^Δ_{c,p, d}$ and $ \mathsf{L}^Δ_{t,p, d}$ such that $$ \big(\mathsf{L}^Δ_{t,p, d}\big)^{-1}\big\|f\big\|_{p}\le \big\|S^Δ(f)\big\|_{p}\le \mathsf{L}^Δ_{c,p, d}\big\|f\big\|_{p},\quad f\in L_p(\mathbb{R}^d). $$ We show that $$\mathsf{L}^Δ_{t,p, d}\approx_d (\mathsf{L}^Δ_{t,p, 1})^d\;\text{ and }\; \mathsf{L}^Δ_{c,p, d}\approx_d (\mathsf{L}^Δ_{c,p, 1})^d.$$ All the previous results can be equally formulated for the $d$-torus $\mathbb{T}^d$. We prove a de Leeuw type transference principle in the vector-valued setting.

math.CA↗
arXiv2021

Notes on real interpolation of operator $L_p$-spaces

Let $\mathcal{M}$ be a semifinite von Neumann algebra. We equip the associated noncommutative $L_p$-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for $1<p<\infty$ let $$L_{p,p}(\mathcal{M})=\big(L_{\infty}(\mathcal{M}),\,L_{1}(\mathcal{M})\big)_{\frac1p,\,p}$$ be equipped with the operator space structure via real interpolation as defined by the second named author ({\em J. Funct. Anal}. 139 (1996), 500--539). We show that $L_{p,p}(\mathcal{M})=L_{p}(\mathcal{M})$ completely isomorphically if and only if $\mathcal{M}$ is finite dimensional. This solves in the negative the three problems left open in the quoted work of the second author. We also show that for $1<p<\infty$ and $1\le q\le\infty$ with $p\neq q$ $$\big(L_{\infty}(\mathcal{M};\ell_q),\,L_{1}(\mathcal{M};\ell_q)\big)_{\frac1p,\,p}=L_p(\mathcal{M}; \ell_q)$$ with equivalent norms, i.e., at the Banach space level if and only if $\mathcal{M}$ is isomorphic, as a Banach space, to a commutative von Neumann algebra. Our third result concerns the following inequality: $$ \big\|\big(\sum_ix_i^q\big)^{\frac1q}\big\|_{L_p(\mathcal{M})}\le\big\|\big(\sum_ix_i^r\big)^{\frac1r}\big\|_{L_p(\mathcal{M})} $$ for any finite sequence $(x_i)\subset L_p^+(\mathcal{M})$, where $0<r<q<\infty$ and $0<p\le\infty$. If $\mathcal{M}$ is not isomorphic, as a Banach space, to a commutative von Meumann algebra, then this inequality holds if and only if $p\ge r$.

math.OA↗
arXiv2021

A Hörmander-Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras

We establish a platform to transfer $L_p$-completely bounded maps on tensor products of von Neumann algebras to $L_p$-completely bounded maps on the corresponding amalgamated free products. As a consequence, we obtain a Hörmander-Mikhlin multiplier theory for free products of groups. Let $\mathbb{F}_\infty$ be a free group on infinite generators $\{g_1, g_2,\cdots\}$. Given $d\ge1$ and a bounded symbol $m$ on $\mathbb{Z}^d$ satisfying the classical Hörmander-Mikhlin condition, the linear map $M_m:\mathbb{C}[\mathbb{F}_\infty]\to \mathbb{C}[\mathbb{F}_\infty]$ defined by $λ(g)\mapsto m(k_1,\cdots, k_d)λ(g)$ for $g=g_{i_1}^{k_1}\cdots g_{i_n}^{k_n}\in\mathbb{F}_\infty$ in reduced form (with $k_l=0$ in $m(k_1,\cdots, k_d)$ for $l>n$), extends to a complete bounded map on $L_p(\widehat{\mathbb{F}}_\infty)$ for all $1<p<\infty$, where $\widehat{\mathbb{F}}_\infty$ is the group von Neumann algebra of $\mathbb{F}_\infty$. A similar result holds for any free product of discrete groups.

math.OA↗
arXiv2020

Atomic decompositions for noncommutative martingales

We prove an atomic type decomposition for the noncommutative martingale Hardy space $\h_p$ for all $0<p<2$ by an explicit constructive method using algebraic atoms as building blocks. Using this elementary construction, we obtain a weak form of the atomic decomposition of $\h_p$ for all $0< p < 1,$ and provide a constructive proof of the atomic decomposition for $p=1$. We also study $(p,\8)_c$-atoms, and show that every $(p,2)_c$-atom can be decomposed into a sum of $(p,\8)_c$-atoms; consequently, for every $0<p\le 1$, the $(p,q)_c$-atoms lead to the same atomic space for all $2\le q\le\8$. As applications, we obtain a characterization of the dual space of the noncommutative martingale Hardy space $\h_p$ ($0<p<1$) as a noncommutative Lipschitz space via the weak form of the atomic decomposition. Our constructive method can also be applied to proving some sharp martingale inequalities.

math.OA↗
arXiv2019

Free Fourier Multipliers associated with the firstSegment

We study Fourier multipliers on free group $\mathbb{F}_\infty$ associated with the first segment of the reduced words, and prove that they are completely bounded on the noncommutative $L^p$ spaces $L^p(\hat{\mathbb{F}}_\infty)$ iff their restriction on $L^p(\hat{\mathbb{F}}_1)=L^p(\mathbb{T})$ are completely bounded. As a consequence, every classical Mikhlin multiplier extends to a $L^p$ Fourier multiplier on free groups for all $1<p<\infty$.

math.OA↗
arXiv2018

Vector-valued Littewood-Paley-Stein theory for semigroups II

Inspired by a recent work of Hytönen and Naor, we solve a problem left open in our previous work joint with Mart\'ınez and Torrea on the vector-valued Littlewood-Paley-Stein theory for symmetric diffusion semigroups. We prove a similar result in the discrete case, namely, for any $T$ which is the square of a symmetric Markovian operator on a measure space $(Ω, μ)$. Moreover, we show that $T\otimes{\rm Id}_X$ extends to an analytic contraction on $L_p(Ω; X)$ for any $1<p<\infty$ and any uniformly convex Banach space $X$.

math.FA↗
arXiv2018

Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori

This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative $d$-torus $\mathbb{T}^d_θ$ (with $θ$ a skew symmetric real $d\times d$-matrix). These spaces share many properties with their classical counterparts. We prove, among other basic properties, the lifting theorem for all these spaces and a Poincaré type inequality for Sobolev spaces. We also show that the Sobolev space $W^k_\infty(\mathbb{T}^d_θ)$ coincides with the Lipschitz space of order $k$, already studied by Weaver in the case $k=1$. We establish the embedding inequalities of all these spaces, including the Besov and Sobolev embedding theorems. We obtain Littlewood-Paley type characterizations for Besov and Triebel-Lizorkin spaces in a general way, as well as the concrete ones in terms of the Poisson, heat semigroups and differences. Some of them are new even in the commutative case, for instance, our Poisson semigroup characterizations improve the classical ones. As a consequence of the characterization of the Besov spaces by differences, we extend to the quantum setting the recent results of Bourgain-Brézis -Mironescu and Maz'ya-Shaposhnikova on the limits of Besov norms. The same characterization implies that the Besov space $B^α_{\infty,\infty}(\mathbb{T}^d_θ)$ for $α>0$ is the quantum analogue of the usual Zygmund class of order $α$. We investigate the interpolation of all these spaces, in particular, determine explicitly the K-functional of the couple $(L_p(\mathbb{T}^d_θ), \, W^k_p(\mathbb{T}^d_θ))$, which is the quantum analogue of a classical result due to Johnen and Scherer. Finally, we show that the completely bounded Fourier multipliers on all these spaces do not depend on the matrix $θ$, so coincide with those on the corresponding spaces on the usual $d$-torus.

math.OA↗
arXiv2017

Noncommutative Davis type decompositions and applications

We prove the noncommutative Davis decomposition for the column Hardy space $\H_p^c$ for all $0<p\leq 1$. A new feature of our Davis decomposition is a simultaneous control of $\H_1^c$ and $\H_q^c$ norms for any noncommutative martingale in $\H_1^c \cap \H_q^c$ when $q\geq 2$. As applications, we show that the Burkholder/Rosenthal inequality holds for bounded martingales in a noncommutative symmetric space associated with a function space $E$ that is either an interpolation of the couple $(L_p, L_2)$ for some $1<p<2$ or is an interpolation of the couple $(L_2, L_q)$ for some $2<q<\infty$. We also obtain the corresponding $Φ$-moment Burkholder/Rosenthal inequality for Orlicz functions that are either $p$-convex and $2$-concave for some $1<p<2$ or are $2$-convex and $q$-concave for some $2<q<\infty$.

math.PR↗
arXiv2016

A noncommutative martingale convexity inequality

Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful semifinite normal weight $ϕ$ and $\mathcal{N}$ be a von Neumann subalgebra of $\mathcal{M}$ such that the restriction of $ϕ$ to $\mathcal{N}$ is semifinite and such that $\mathcal{N}$ is invariant by the modular group of $ϕ$. Let $\mathcal{E}$ be the weight preserving conditional expectation from $\mathcal{M}$ onto $\mathcal{N}$. We prove the following inequality: \[\|x\|_p^2\ge\bigl \|\mathcal{E}(x)\bigr\|_p^2+(p-1)\bigl\|x-\mathcal{E}(x)\bigr\|_p^2, \qquad x\in L_p(\mathcal{M}),1 0$ such that for any free group $\mathbb{F}_n$ and any $q\ge4-\varepsilon_0$, \[\|P_t\|_{2\to q}\le1\quad\Leftrightarrow\quad t\ge\log{\sqrt{q-1}},\] where $(P_t)$ is the Poisson semigroup defined by the natural length function of $ \mathbb{F}_n$.

math.OA↗
arXiv2016

Khintchine type inequalities for reduced free products and Applications

We prove Khintchine type inequalities for words of a fixed length in a reduced free product of $C^*$-algebras (or von Neumann algebras). These inequalities imply that the natural projection from a reduced free product onto the subspace generated by the words of a fixed length $d$ is completely bounded with norm depending linearly on $d$. We then apply these results to various approximation properties on reduced free products. As a first application, we give a quick proof of Dykema's theorem on the stability of exactness under the reduced free product for $C^*$-algebras. We next study the stability of the completely contractive approximation property (CCAP) under reduced free product. Our first result in this direction is that a reduced free product of finite dimensional $C^*$-algebras has the CCAP. The second one asserts that a von Neumann reduced free product of injective von Neumann algebras has the weak-$*$ CCAP. In the case of group $C^*$-algebras, we show that a free product of weakly amenable groups with constant 1 is weakly amenable.

math.OA↗
arXiv2016

Characterizations of operator-valued Hardy spaces and applications to harmonic analysis on quantum tori

This paper deals with the operator-valued Hardy spaces introduced and studied by Tao Mei. Our principal result shows that the Poisson kernel in Mei's definition of these spaces can be replaced by any reasonable test function. As an application, we get a general characterization of Hardy spaces on quantum tori. The latter characterization plays a key role in our recent study of Triebel-Lizorkin spaces on quantum tori.

math.OA↗
arXiv2014

$H^\infty$ functional calculus and maximal inequalities for semigroups of contractions on vector-valued $L_p$-spaces

Let $\{T_t\}_{t>0}$ be a strongly continuous semigroup of positive contractions on $L_p(X,μ)$ with $1 0}\frac1t\Big|\int_0^tT_s(f(\cdot,ω))(x)ds\Big|,\quad (x,ω)\in X\timesΩ.$$ Then the following maximal ergodic inequality holds $$\big\|\mathcal M(f)\big\|_{L_p(X; E)}\lesssim \big\|f\big\|_{L_p(X; E)},\quad f\in L_p(X; E).$$ If the semigroup $\{T_t\}_{t>0}$ is additionally assumed to be analytic, then $\{T_t\}_{t>0}$ extends to an analytic semigroup on $L_p(X; E)$ and $\mathcal M(f)$ in the above inequality can be replaced by the following sectorial maximal function $$\mathcal T_θ(f)(x, ω)=\sup_{|{\rm arg}(z)|<θ}\big|T_z(f(\cdot,ω))(x)\big|$$ for some $θ>0$. Under the latter analyticity assumption and if $E$ is a complex interpolation space between a Hilbert space and a UMD Banach space, then $\{T_t\}_{t>0}$ extends to an analytic semigroup on $L_p(X; E)$ and its negative generator has a bounded $H^\infty(Σ_σ)$ calculus for some $σ<π/2$.

math.FA↗
arXiv2013

Weighted variation inequalities for differential operators and singular integrals

We prove weighted strong $q$-variation inequalities with $2<q<\infty$ for differential and singular integral operators. For the first family of operators the weights used can be either Sawyer's one-sided $A^+_p$ weights or Muckenhoupt's $A_p$ weights according to that the differential operators in consideration are one-sided or symmetric. We use only Muckenhoupt's $A_p$ weights for the second family. All these inequalities hold equally in the vector-valued case, that is, for functions with values in $\el^ρ$ for $1<ρ<\infty$. As application, we show variation inequalities for mean bounded positive invertible operators on $L^p$ with positive inverses.

math.CA↗