A Markov dilation for self-adjoint Schur multipliers

Completely compact Herz-Schur multipliers of dynamical systems

We prove that if $G$ is a discrete group and $(A,G,α)$ is a C*-dynamical system such that the reduced crossed product $A\rtimes_{r,α} G$ possesses property (SOAP) then every completely compact Herz-Schur $(A,G,α)$-multiplier can be approximated in the completely bounded norm by Herz-Schur $(A,G,α)$-multipliers of finite rank. As a consequence, if $G$ has the approximation property (AP) then the completely compact Herz-Schur multipliers of $A(G)$ coincide with the closure of $A(G)$ in the completely bounded multiplier norm. We study the class of invariant completely compact Herz-Schur multiplie…

Schur multipliers and operator-valued Foguel-Hankel operators

We show that some matrices are Schur multipliers and this is applied to obtain classes of operator-valued Foguel-Hankel operators similar to contractions. This provides partial answers to a problem of K. Davidson and the second author concerning CAR-valued Foguel-Hankel operators.

Unconditionality, Fourier multipliers and Schur multipliers

Let $G$ be an infinite locally compact abelian group. If $X$ is Banach space, we show that if every bounded Fourier multiplier $T$ on $L^2(G)$ has the property that $T\ot Id_X$ is bounded on $L^2(G,X)$ then the Banach space $X$ is isomorphic to a Hilbert space. Moreover, if $1<p<\infty$, $p\not=2$, we prove that there exists a bounded Fourier multiplier on $L^p(G)$ which is not completely bounded. Finally, we examine unconditionality from the point of view of Schur multipliers. More precisely, we give several necessary and sufficient conditions to determine if an operator space is completely i…

Schur multipliers and de Branges-Rovnyak spaces: the multiscale case

We consider bounded linear operators acting on the $\ell_2$ space indexed by the nodes of a homogeneous tree. Using the Cuntz relations between the primitive shifts on the tree, we generalize the notion of the single-scale time-varying point evaluation and introduce the corresponding reproducing kernel Hilbert space in which Cauchy's formula holds. These notions are then used in the study of the Schur multipliers and of the associated de Branges-Rovnyak spaces. As an application we obtain realization of Schur multipliers as transfer operators of multiscale input-state-output systems.

Schur and Fourier multipliers of an amenable group acting on non-commutative Lp-spaces

Consider a completely bounded Fourier multiplier phi of a locally compact group G, and take 1 <= p <= infinity. One can associate to phi a Schur multiplier on the Schatten classes S_p(L^2 G), as well as a Fourier multiplier on Lp(LG), the non-commutative Lp-space of the group von Neumann algebra of G. We prove that the completely bounded norm of the Schur multiplier is not greater than the completely bounded norm of the Lp-Fourier multiplier. When G is amenable we show that equality holds, extending a result by Neuwirth and Ricard to non-discrete groups. For a discrete group G and in the speci…

作用于非幺模群非交换 $L_p$-空间上的多重线性 Fourier 与 Schur 乘子的转移性质

Transference of multilinear Fourier and Schur multipliers acting on non-commutative $L_p$-spaces for non-unimodular groups

本文推广了非幺模群上多重线性 Fourier 乘子的定义，并证明了在非交换 $L_p$-空间上，Fourier 与 Schur 乘子间的转移结果依然成立。

Markov dilations of semigroups of Fourier multipliers

We describe a Markov dilation for any weak* continuous semigroup $(T_t)_{t \geq 0}$ of selfadjoint unital completely positive Fourier multipliers acting on the group von Neumann algebra $\mathrm{VN}(G)$ of a locally compact group $G$.

Bilinear forms, Schur multipliers, complete boundedness and duality

Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex m x n matrices. Based on the theory of operator spaces and completely bounded mappings we present norm optimal versions of these results and two norm optimal factorization results related to the Schur product. We show that the spaces of respectively bilinear forms and Schur multipliers are conjugate duals to each other with respect to their completely bounded norms.

Schur乘积范数及其对偶范数

The Schur multiplier norm and its dual norm

本文研究了复自伴矩阵的Schur乘积范数及其对偶范数的计算公式，给出了自伴矩阵的Schur乘积范数和对偶范数的定义及性质。

幂等Schur乘子的局部几何

The local geometry of idempotent Schur multipliers

本文研究幂等Schur乘子，利用边界平坦性概念刻画其在Schatten $p$-类上的有界性，证明其均以三角投影为模型，并推广至Lie群上Fourier幂等元的局部$L_p$-有界性，是Fefferman球乘子定理的非交换推广。

树上的径向Schur乘子与Fourier-Schur转移的失效

Radial Schur multipliers on trees and the failure of Fourier-Schur transference

该研究刻画了树上Schatten p类的径向Schur乘子，证明其算子范数与环面Fourier乘子范数的不等式，构造自由群中完全S_p有界但Fourier乘子无界的Herz-Schur乘子，否定了Fourier-Schur转移。

Compactness of Schur-A Multipliers and Haagerup Tensorproduct

In this paper we study the connection between Haagerup tensor product and compactness of Schur $A$-multiplier. In particular, we give a new characterization of elementary $C^{\ast}$-algebra in terms of completely compact Schur $A$-multiplier.

Riesz-Schur transforms

We investigate nontrigonometric forms of Riesz transforms in the context of Schur multipliers. This refines Grothendieck-Haagerup's endpoint criterion with a new condition for the Schatten p-boundedness of Schur multipliers and strengthens Potapov/Sukochev's solution of Arazy's conjecture. We recover as well dimension-free estimates for trigonometric Riesz transforms. Our discrete approach is much simpler than previous harmonic analysis and probabilistic approaches. As an application, we find a very simple proof of recent criteria for Schur multipliers of Hörmander-Mikhlin and Marcinkiewicz ty…

Schur multipliers of Cartan pairs

We define the Schur multipliers of a separable von Neumann algebra M with Cartan masa A, generalising the classical Schur multipliers of $B(\ell^2)$. We characterise these as the normal A-bimodule maps on M. If M contains a direct summand isomorphic to the hyperfinite II_1 factor, then we show that the Schur multipliers arising from the extended Haagerup tensor product $A \otimes_{eh} A$ are strictly contained in the algebra of all Schur multipliers.

Schur Multipliers and Matrix Products

We give necessary and sufficient conditions for a Schur map to be a homomorphism, with some generalizations to the infinite-dimensional case. In the finite-dimensional case, we find that a Schur multiplier distributes over matrix multiplication if and only if the Schur matrix has a certain simple form.

Measurable Schur Multipliers and Completely Bounded Multipliers of the Fourier Algebras

Let G be a locally compact group L^p(G) be the usual L^p-space for 1 =< p =< infty and A(G) be the Fourier algebra of G. Our goal is to study, in a new abstract context, the completely bounded multipliers of A(G), which we denote M_{cb}A(G). We show that M_{cb}A(G) can be characterised as the ``invariant part'' of the space of (completely) bounded normal L^{infty}(G)-bimodule maps on B(L^2(G)), the space of bounded operator on L^2(G). In doing this we develop a function theoretic description of the normal L^{infty}(X,mu)-bimodule maps on B(L^2(X,mu)), which we denote by V^{infty}(X,μ), and nam…

Positive extensions of Schur multipliers

We introduce partially defined Schur multipliers and obtain necessary and sufficient conditions for the existence of extensions to fully defined positive Schur multipliers, in terms of operator systems canonically associated with their domains. We use these results to study the problem of extending a positive definite function defined on a symmetric subset of a locally compact group to a positive definite function defined on the whole group.

Norms of Schur Multipliers

A subset P of N x N is called Schur bounded if every infinite matrix with bounded entries which is zero off of P yields a bounded Schur multiplier on B(H). Such sets are characterized as being the union of a subset with at most k entries in each row with another that has at most k entries in each column, for some finite k. If k is optimal, there is a Schur multiplier supported on the pattern with norm O(k^(1/2)), which is sharp up to a constant. The same techniques give a new, more elementary proof of results of Varopoulos and Pisier on Schur multipliers with given matrix entries of random sig…

Exactness and SOAP of Crossed Products via Herz--Schur multipliers

Given a $C^*$-dynamical system $(A,G,α)$, with $G$ a discrete group, Schur $A$-multipliers and Herz--Schur $(A,G,α)$-multipliers are used to implement approximation properties, namely exactness and the strong operator approximation property (SOAP), of $A \rtimes_{α, r} G$. The resulting characterisations of exactness and SOAP of $A \rtimes_{α, r} G$ generalise the corresponding statements for the reduced group $C^*$-algebra.

Multilinear transference of Fourier and Schur multipliers acting on non-commutative $L_p$-spaces

Let $G$ be a locally compact unimodular group, and let $ϕ$ be some function of $n$ variables on $G$. To such a $ϕ$, one can associate a multilinear Fourier multiplier, which acts on some $n$-fold product of the non-commutative $L_p$-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an $n$-fold product of Schatten classes $S_p(L_2(G))$. We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called `multiplicatively bounded $(p_1,\ldots,p_n)$-norm' of a multilinear Sch…

Multipliers and Duality for Group Actions

We define operator-valued Schur and Herz--Schur multipliers in terms of module actions, and show that the standard properties of these multipliers follow from well-known facts about these module actions and duality theory for group actions. These results are applied to study the Herz--Schur multipliers of an abelian group acting on its Pontryagin dual: it is shown that a natural subset of these Herz--Schur multipliers can be identified with the classical Herz--Schur multipliers of the direct product of the group with its dual group.

Local Operator Multipliers and Positivity

We establish an unbounded version of Stinespring's Theorem and a lifting result for Stinespring representations of completely positive modular maps defined on the space of all compact operators. We apply these results to study positivity for Schur multipliers. We characterise positive local Schur multipliers, and provide a description of positive local Schur multipliers of Toeplitz type. We introduce local operator multipliers as a non-commutative analogue of local Schur multipliers, and obtain a characterisation that extends earlier results concerning operator multipliers and local Schur mult…

Semigroups of Herz-Schur Multipliers

In order to investigate the relationship between weak amenability and the Haagerup property for groups, we introduce the weak Haagerup property, and we prove that having this approximation property is equivalent to the existence of a semigroup of Herz-Schur multipliers generated by a proper function. It is then shown that a (not necessarily proper) generator of a semigroup of Herz-Schur multipliers splits into a positive definite kernel and a conditionally negative definite kernel. We also show that the generator has a particularly pleasant form if and only if the group is amenable. In the sec…

Multiplier algebras of complete Nevanlinna-Pick spaces: dilations, boundary representations and hyperrigidity

We study reproducing kernel Hilbert spaces on the unit ball with the complete Nevanlinna-Pick property through the representation theory of their algebras of multipliers. We give a complete description of the representations in terms of the reproducing kernels. While representations always dilate to $*$-representations of the ambient $C^*$-algebra, we show that in our setting we automatically obtain coextensions. In fact, we show that in many cases, this phenomenon characterizes the complete Nevanlinna-Pick property. We also deduce operator theoretic dilation results which are in the spirit of…

Radial Schur multipliers on some generalisations of trees

We give a characterisation of radial Schur multipliers on finite products of trees. The equivalent condition is that a certain generalised Hankel matrix involving the discrete derivatives of the radial function is a trace class operator. This extends Haagerup, Steenstrup and Szwarc's result for trees. The same condition can be expressed in terms of Besov spaces on the torus. We also prove a similar result for products of hyperbolic graphs and provide a sufficient condition for a function to define a radial Schur multiplier on a finite dimensional CAT(0) cube complex.

Herz-Schur Multipliers of Fell Bundles

In this paper we develop the notion of Herz-Schur multipliers to the context of Fell bundle, and we give a necessary and sufficient condition if the reduced cross-sectional $C^{\ast}$-algebra $C_r^{\ast}(\mathfrak{B})$ of Fell bundle $\mathfrak{B}$ over discrete group $G$ is nuclear in terms of this generalized notion. As an application, we prove that if $C_r^{\ast}(\mathfrak{B})$ is nuclear, then for any subgroup $H \subset G$, the $C^{\ast}$-algebra $C_r^{\ast}(\mathfrak{B}_H)$ is nuclear.

Dilations of markovian semigroups of Fourier multipliers on locally compact groups

We prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of Markov Fourier multipliers acting on a group von Neumann algebra $\mathrm{VN}(G)$ associated to a locally compact group $G$ can be dilated by a weak* continuous group of Markov $*$-automorphisms on a bigger von Neumann algebra. Our construction relies on probabilistic tools and is even new for the group $\mathbb{R}^n$. Our results imply the boundedness of the McIntosh's $\mathrm{H}^\infty$ functional calculus of the generators of these semigroups on the associated noncommutative $\mathrm{L}^p$-spaces.

S₄上算子范数与完全有界范数不等的Schur乘子

Schur Multipliers with Unequal Operator and Completely Bounded Norms on $S_p$, $1<p\ne 2<\infty $

该研究构造S₄上显式Schur乘子，证明其算子范数小于完全有界范数，回答了相关问题并否定特定论断，还得到包含4的开区间I使对应不等式成立。

Positive Herz-Schur multipliers and approximation properties of crossed products

For a $C^*$-algebra $A$ and a set $X$ we give a Stinespring-type characterisation of the completely positive Schur $A$-multipliers on $K(\ell^2(X))\otimes A$. We then relate them to completely positive Herz-Schur multipliers on $C^*$-algebraic crossed products of the form $A\rtimes_{α,r} G$, with $G$ a discrete group, whose various versions were considered earlier by Anantharaman-Delaroche, Bédos and Conti, and Dong and Ruan. The latter maps are shown to implement approximation properties, such as nuclearity or the Haagerup property, for $A\rtimes_{α,r} G$.

舒尔乘子在调和分析与算子代数中的影响

The impact of Schur multipliers in harmonic analysis and operator algebras

本文综述舒尔乘子在过去15年的影响，关注其与傅里叶乘子的联系及在调和分析和算子代数间的桥梁作用，介绍舒尔滕p类最新界在相关领域的应用，以及非托普利茨舒尔乘子等方面的新进展。

与双线性形式和 Schur 乘子相关的唯一矩阵分解

Unique Matrix Factorizations associated to Bilinear Forms and Schur Multipliers

本文利用算子空间理论研究复 m×n 矩阵的 Grothendieck 型范数最优分解，证明四类分解中三类唯一、一类在特定条件下唯一。

Operator system structures and extensions of Schur multipliers

For a given C*-algebra $\mathcal{A}$, we establish the existence of maximal and minimal operator $\mathcal{A}$-system structures on an AOU $\mathcal{A}$-space. In the case $\mathcal{A}$ is a W*-algebra, we provide an abstract characterisation of dual operator $\mathcal{A}$-systems, and study the maximal and minimal dual operator $\mathcal{A}$-system structures on a dual AOU $\mathcal{A}$-space. We introduce operator-valued Schur multipliers, and provide a Grothendieck-type characterisation. We study the positive extension problem for a partially defined operator-valued Schur multiplier $φ$ and…

Separating Fourier and Schur multipliers

Let $G$ be a locally compact unimodular group, let $1\leq p<\infty$,let $ϕ\in L^\infty(G)$ and assume that the Fourier multiplier $M_ϕ$associated with $ϕ$ is bounded on the noncommutative $L^p$-space $L^p(VN(G))$.Then $M_ϕ\colon L^p(VN(G))\to L^p(VN(G))$ is separating (that is,$\{a^*b=ab^*=0\}\Rightarrow\{M_ϕ(a)^* M_ϕ(b)=M_ϕ(a)M_ϕ(b)^*=0\}$for any $a,b\in L^p(VN(G))$) if and only if thereexists $c\in\mathbb C$ and a continuouscharacter $ψ\colon G\to\mathbb C$ such that $ϕ=cψ$ locally almost everywhere. This provides a characterization of isometricFourier multipliers on $L^p(VN(G))$, when $p\no…

Multidimensional operator multipliers

We introduce multidimensional Schur multipliers and characterise them generalising well known results by Grothendieck and Peller. We define a multidimensional version of the two dimensional operator multipliers studied recently by Kissin and Shulman. The multidimensional operator multipliers are defined as elements of the minimal tensor product of several C*-algebras satisfying certain boundedness conditions. In the case of commutative C*-algebras, the multidimensional operator multipliers reduce to continuous multidimensional Schur multipliers. We show that the multipliers with respect to som…

傅里叶乘子的绝对膨胀

Absolute dilation of Fourier multipliers

本文研究冯·诺依曼代数上傅里叶乘子的绝对膨胀问题，建立了离散顺从群上的转移定理，并据此刻画了具有绝对膨胀的傅里叶乘子，同时构造了对称群S3中首个不允许绝对膨胀的单位完全正傅里叶乘子反例，并证明所有阿贝尔群上的傅里叶乘子均允许绝对膨胀。

Herz-Schur Multipliers and Non-Uniformly Bounded Representations of Locally Compact Groups

Let G be a second countable, locally compact group and let f be a continuous Herz-Schur multiplier on G. Our main result gives the existence of a (not necessarily uniformly bounded) strongly continuous representation on a Hilbert space, such that f is the coefficient of this representation with respect to two vectors with bounded orbit. Moreover, we show that the norm of the representation of an element g from G is at most exponential in terms of the metric distance from g to the identity element of G.

Schur Multipliers and Spherical Functions on Homogeneous Trees

Let X be a homogeneous tree of degree q+1 (for q between 2 and infinity) and let f be a complex function on X times X for which f(x,y) only depend on the distance between x and y in X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X times X. Moreover, we find a closed expression for the Schur norm of f. As applications, we obtain a closed expression for the completely bounded Fourier multiplier norm of the radial functions on the free (non-abelian) group on N generators (for N between 2 and infinity) and of the spherical functions on…

Schatten-von Neumann 类中的 Schur 乘子

Schur multipliers in Schatten-von Neumann classes

本文提出了Schatten p-类上Schur乘子有界性的新准则，解决了de la Salle猜想，推广了Hörmander-Mikhlin定理，并证实了Arazy猜想，进而导出S_p-范数的Littlewood-Paley刻画及在调和分析中的应用。

通过准局域性和Schur乘子刻画粗嵌入性质

Characterisations for coarse embeddability via quasi-locality and Schur multipliers

本研究针对有界几何的离散度量空间，通过准局域性、Schur乘子等建立其粗嵌入到希尔伯特空间的新刻画，还探讨了相关应用及与ℓ^p嵌入的关系。

Multipliers of the Hardy space H^1 and power bounded operators

We study the space of functions $ϕ\colon \NN\to \CC$ such that there is a Hilbert space $H$, a power bounded operator $T$ in $B(H)$ and vectors $ξ,η$ in $H$ such that $$ϕ(n) = < T^nξ,η>.$$ This implies that the matrix $(ϕ(i+j))_{i,j\ge 0}$ is a Schur multiplier of $B(\ell_2)$ or equivalently is in the space $(\ell_1 \buildrel {\vee}\over {\otimes} \ell_1)^*$. We show that the converse does not hold, which answers a question raised by Peller [Pe]. Our approach makes use of a new class of Fourier multipliers of $H^1$ which we call ``shift-bounded''. We show that there is a $ϕ$ which is a ``compl…

Arazy关于Schur乘子的猜想：重访与解决

Arazy's conjecture concerning Schur multipliers: revisited and resolved

本文解决了Arazy关于Schur乘子有界性的猜想，确定了Schur-Hadamard乘子从$\mathcal{S}^q$到$\mathcal{S}^p$有界的精确条件，并给出了新证明。

绝对可膨胀 Schur 乘子的一个刻画

A characterization of absolutely dilatable Schur multipliers

本文在配备正规半有限忠实迹的 von Neumann 代数框架下，通过引入绝对可膨胀概念，刻画了使得 Schur 乘法算子具备该性质的 $σ$-有限测度空间上的有界 Schur 乘子，并在可分情形下给出了基于 von Neumann 代数与迹的等价条件。

$σ$-有限 von Neumann 代数的 BMO 空间与 $SU_q(2)$ 上的 Fourier-Schur 乘子

BMO spaces of $σ$-finite von Neumann algebras and Fourier-Schur multipliers on $SU_q(2)$

本文研究 $σ$-有限 von Neumann 代数上的半群 BMO 空间，证明其行列空间具有预对偶、为 Banach 空间并满足插值公式，同时引入紧量子群上的 Fourier-Schur 乘子并在 $SU_q(2)$ 上构造实例。

关于二阶差商函数 Schur 乘子的最佳常数

On the best constants of Schur multipliers of second order divided difference functions

本文基于双线性转移和Hörmander-Mikhlin-Schur乘子定理，给出了二阶差商函数双线性Schur乘子有界性的新证明，显著改进了其渐近界的上下界并证明了其在特定条件下的最优性。

Approximation properties for noncommutative Lp-spaces associated with lattices in Lie groups

In 2010, Lafforgue and de la Salle gave examples of noncommutative Lp-spaces without the operator space approximation property (OAP) and, hence, without the completely bounded approximation property (CBAP). To this purpose, they introduced the property of completely bounded approximation by Schur multipliers on Sp and proved that for p < 4/3 and p > 4 the groups SL(n,Z), with n \geq 3, do not have it. Since for 1 < p < \infty the property of completely bounded approximation by Schur multipliers on Sp is weaker than the approximation property of Haagerup and Kraus (AP), these groups were also t…

Schur multipliers and mixed unitary maps

We consider the tensor product of the completely depolarising channel on $d\times d$ matrices with the map of Schur multiplication by a $k \times k$ correlation matrix and characterise, via matrix theory methods, when such a map is a mixed (random) unitary channel. When $d=1$, this recovers a result of O'Meara and Pereira, and for larger $d$ is equivalent to a result of Haagerup and Musat that was originally obtained via the theory of factorisation through von Neumann algebras. We obtain a bound on the distance between a given correlation matrix for which this tensor product is nearly mixed un…

$C^*$-代数的Schur乘子、群不变紧化及其在顺从性和渗流中的应用

Schur Multipliers of $C^*$-algebras, group-invariant compactification and applications to amenability and percolation

本文通过构造约化群C*-代数上Schur乘子的正交分解，利用群不变紧化中的极限点，给出离散群顺从性的变分与渗流两种新刻画，揭示Schur乘子与群几何性质的深层联系。

关于更高阶除差函数Schur乘子的最佳常数

On the best constants of Schur multipliers of higher order divided difference functions

本文研究了更高阶除差函数Schur乘子的最佳常数，通过分析多元Schur乘子的范数，得出了关于Hölder共轭和函数导数的估计，提供了Koplienko问题解决的关键定理的另一种证明。

ucp自伴Fourier乘子的绝对膨胀：非幺模情形

Absolute dilations of ucp self-adjoint Fourier multipliers: the non unimodular case

本文证明非幺模局部紧群上由实值函数诱导的单位完全正自伴Fourier乘子具有绝对膨胀，进而其L^p实现可膨胀为等距，且当函数取值于[0,1]时为Ritt算子。