We extend the notion of Herz-Schur multipliers to the setting of non-commutative dynamical systems: given a C*-algebra $A$, a locally compact group $G$, and an action $α$ of $G$ on $A$, we define transformations on the (reduced) crossed product $A\rtimes_{r,α} G$ of $A$ by $G$, which, in the case $A = \mathbb{C}$, reduce to the classical Herz-Schur multipliers. We also introduce a class of Schur $A$-multipliers, establish its characterisation which generalise the classical descriptions of Schur multipliers and present a transference theorem in the new setting, identifying isometrically the Herz-Schur multipliers of the dynamical system $(A,G,α)$ with the invariant part of the Schur $A$-multipliers. We discuss special classes of Herz-Schur multipliers, in particular, those which are associated to a locally compact abelian group $G$ and its canonical action on the $C^*$-algebra $C^*(Γ)$ of the dual group $Γ$.
We obtain descriptions of central operator-valued Schur and Herz-Schur multipliers, akin to a classical characterisation due to Grothendieck, that reveals a close link between central (linear) multipliers and bilinear multipliers into the trace class. Restricting to dynamical systems where a locally compact group acts on itself by translation, we identify their convolution multipliers as the right completely bounded multipliers, in the sense of Junge-Neufang-Ruan, of a canonical quantum group associated with the underlying group. We provide characterisations of contractive idempotent operator-…
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…
We introduce a noncommutative analogue of the Figá-Talamanca-Herz algebra $A_p(G)$ on the natural predual of the operator space $\frak{M}_{p,cb}$ of completely bounded Schur multipliers on Schatten space $S_p$. We determine the isometric Schur multipliers and prove that the space $\frak{M}_{p}$ of bounded Schur multipliers on Schatten space $S_p$ is the closure in the weak operator topology of the span of isometric 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…
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.
Let $G$ be a locally compact group and $p,q\in \mathbb{R}$ with $p>1$ $p\not=2$ and $q$ between $2$ and $p$ (if $p<2$ then $p<q<2$, if $p>2$ then $2<q<p.$) The main result of the paper is that $A_q(G)$ multiplies $A_p(G)$, more precisely we show that the Banach algebra $A_p(G)$is a Banach module on $A_q(G).$
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.
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.
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 $A\rtimes_{α,r} G$.
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.
A linear map $u\colon \ E\to F$ between operator spaces is called completely co-bounded if it is completely bounded as a map from $E$ to the opposite of $F$. We give several simple results about completely co-bounded Schur multipliers on $B(\ell_2)$ and the Schatten class $S_p$. We also consider Herz-Schur multipliers on groups.
Schur multipliers were introduced by Schur in the early 20th century and have since then found a considerable number of applications in Analysis and enjoyed an intensive development. Apart from the beauty of the subject in itself, sources of interest in them were connections with Perturbation Theory, Harmonic Analysis, the Theory of Operator Integrals and others. Advances in the quantisation of Schur multipliers were recently made by Kissin and Shulman. The aim of the present article is to summarise a part of the ideas and results in the theory of Schur and operator multipliers. We start with …
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.
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…
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.
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…
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.
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…
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.
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.
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…
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…
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.
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.
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…
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…
Using the recently developed notion of a Herz--Schur multiplier of a C*-dynamical system we introduce weak amenability of C*- and W*-dynamical systems. As a special case we recover Haagerup's characterisation of weak amenability of a discrete group. We also consider a generalisation of the Fourier algebra to crossed products and study its multipliers.
Schur Multipliers with Unequal Operator and Completely Bounded Norms on $S_p$, $1<p\ne 2<\infty $
Institute for Advanced Study in Mathematics of HIT, Harbin Institute of Technology（哈尔滨工业大学数学高等研究院）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；Department of Mathematics, New Uzbekistan University（新乌兹别克斯坦大学数学系）；V. I. Romanovsky Institute of Mathematics of the Academy of Sciences of Uzbekistan（乌兹别克斯坦科学院V.I.罗曼诺夫斯基数学研究所）
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.
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…
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…
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…
Given a collection of test functions, one defines the associated Schur-Agler class as the intersection of the contractive multipliers over the collection of all positive kernels for which each test function is a contractive multiplier. We indicate extensions of this framework to the case where the test functions, kernel functions, and Schur-Agler-class functions are allowed to be matrix- or operator-valued. We illustrate the general theory with two examples: (1) the matrix-valued Schur class over a finitely-connected planar domain and (2) the matrix-valued version of the constrained Hardy alge…
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…
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…
Characterisations for coarse embeddability via quasi-locality and Schur multipliers
Centre for Mathematical Sciences, Lund University（隆德大学数学中心）；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）；College of Data Science, Jiaxing University（嘉兴学院数据科学学院）
We establish several new characterizations of amenable $W^*$- and $C^*$-dynamical systems over arbitrary locally compact groups. In the $W^*$-setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz-Schur multipliers of $(M,G,α)$ converging point weak* to the identity of $G\bar{\ltimes}M$. In the $C^*$-setting, we prove that amenability of $(A,G,α)$ is equivalent to an analogous Herz-Schur multiplier approximation of the identity of the reduced crossed product $G\ltimes A$, as well as a particular case of the pos…
Arazy's conjecture concerning Schur multipliers: revisited and resolved
Institute for Advanced Study in Mathematics of HIT, Harbin Institute of Technology（哈尔滨工业大学数学高等研究院）；School of Mathematics and Statistics, University of NSW（新南威尔士大学数学与统计学院）
We show that the multiplier algebra of the Fourier algebra on a locally compact group $G$ can be isometrically represented on a direct sum on non-commutative $L^p$ spaces associated to the right von Neumann algebra of $G$. If these spaces are given their canonical Operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the non-commutative $L^p$ spaces we construct, and show that they are completely isometric to those considered recently by Forrest, Lee and Samei; we improve a result about module homomo…
We study a generalisation of operator spaces modelled on $L_p$ spaces, instead of Hilbert spaces, using the notion of $p$-complete boundedness, as studied by Pisier and Le Merdy. We show that the Figá-Talamanca-Herz Algebras $A_p(G)$ becomes quantised Banach algebras in this framework, and that the cohomological notion of amenability of these algebras corresponds to amenability of the locally compact group $G$. We thus argue that we have presented a generalised of the use of operator spaces in studying the Fourier algebra $A(G)$, in the spirit of Ruan. Finally, we show that various notions of …
Let A_i be a family of unital C*-algebras, respectively, of von Neumann algebras and phi: N_0 \to C. We show that if a Hankel matrix related to phi is trace-class, then there exists a unique completely bounded map M_phi on the reduced free product of the A_i, which acts as an radial multiplier. Hereby we generalize a result of Wysoczański for Herz-Schur multipliers on reduced group C*-algebras for free products of groups.
We introduce a noncommutative analogue of the absolute value of a regular operator acting on a noncommutative $\mathrm{L}^p$-space. We equally prove that two classical operator norms, the regular norm and the decomposable norm are identical. We also describe precisely the regular norm of several classes of regular multipliers. This includes Schur multipliers and Fourier multipliers on some unimodular locally compact groups which can be approximated by discrete groups in various senses. A main ingredient is to show the existence of a bounded projection from the space of completely bounded $\mat…
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…
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…
We define a sequence of functions, namely tame cuts, in the Fourier algebra $A(G)$ of a locally compact group $G$, that satisfies certain convergence and growth conditions. This new consideration allows us to give a group admitting a Fourier multiplier that is not completely bounded. Furthermore, we show that the induction map $MA(Γ)\rightarrow MA(G)$ is not always continuous. We also show how Liao's Property $(T_{Schur}, G, K)$ opposes tame cuts. Some examples are provided.
