Let $\mathbb{G}$ be a locally compact quantum group with dual $\widehat{\mathbb{G}}$. Suppose that the left Haar weight $φ$ and the dual left Haar weight $\widehatφ$ are tracial, e.g. $\mathbb{G}$ is a unimodular Kac algebra. We prove that for $1<p\le 2 \le q<\infty$, the Fourier multiplier $m_{x}$ is bounded from $L_p(\widehat{\mathbb{G}},\widehatφ)$ to $L_q(\widehat{\mathbb{G}},\widehatφ)$ whenever the symbol $x$ lies in $L_{r,\infty}(\mathbb{G},φ)$, where $1/r=1/p-1/q$. Moreover, we have
\begin{equation*}
\|m_{x}:L_p(\widehat{\mathbb{G}},\widehatφ)\to L_q(\widehat{\mathbb{G}},\widehatφ)\|\le c_{p,q} \|x\|_{L_{r,\infty}(\mathbb{G},φ)},
\end{equation*}
where $c_{p,q}$ is a constant depending only on $p$ and $q$. This was first proved by Hörmander \cite{Hormander1960} for $\mathbb{R}^n$, and was recently extended to more general groups and quantum groups. Our work covers all these results and the proof is simpler. In particular, this also yields a family of $L_p$-Fourier multipliers over discrete group von Neumann algebras. A similar result for $\mathcal{S}_p$-$\mathcal{S}_q$ Schur multipliers is also proved.
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 $1<p<\infty$.
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…
In this paper we establish the $L^p$-$L^q$ boundedness of Fourier multipliers on locally compact separable unimodular groups for the range of indices $1<p\leq 2 \leq q<\infty$. Our approach is based on the operator algebras techniques. The result depends on a version of the Hausdorff-Young-Paley inequality that we establish on general locally compact separable unimodular groups. In particular, the obtained result implies the corresponding Hörmander's Fourier multiplier theorem on $\mathbb{R}^{n}$ and the corresponding known results for Fourier multipliers on compact Lie groups.
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…
In this paper, we investigate the structure of the multiplier module of a Hilbert module over a locally C*-algebra and the relationship between the set of all adjointable operators from a Hilbert A-module E to a Hilbert A-module F and the set of all adjointable operators from the multiplier module M(E)of E to the multiplier module M(F) of F.
We investigate Fourier multipliers on the compact dual of arbitrary discrete groups. Our main result is a Hörmander-Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition. We also find Littlewood-Paley type inequalities in group von Neumann algebras, prove $L_p$ estimates for noncommutative Riesz transforms and characterize $L_\infty \to \mathrm{BMO}$ boundedness for radial Fourier multipliers. The key novelties of our approach are to exploit group cocycles and cross products in Fourier multiplier theory in conjunction with BMO spaces associated to semigrou…
We study multiplier algebras for a large class of Banach algebras which contains the group algebra $L_1(G)$, the Beurling algebras $L_1(G, ω)$, and the Fourier algebra $A(G)$ of a locally compact group $G$. This study yields numerous new results and unifies some existing theorems on $L_1(G)$ and $A(G)$ through an abstract Banach algebraic approach. Applications are obtained on representations of multipliers over locally compact quantum groups and on topological centre problems. In particular, five open problems in abstract harmonic analysis are solved.
Let $G$ be a locally compact group, and let $A_\cb(G)$ denote the closure of $A(G)$, the Fourier algebra of $G$, in the space of completely bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group such that $\cstar(G)$ is residually finite-dimensional, we show that $A_\cb(G)$ is operator amenable. In particular, $A_\cb(F_2)$ is operator amenable even though $F_2$, the free group in two generators, is not an amenable group. Moreover, we show that, if $G$ is a discrete group such that $A_\cb(G)$ is operator amenable, a closed ideal of $A(G)$ is weakly completely complemented in…
We investigate Fourier multipliers with smooth symbols defined over locally compact Hausdorff groups. Our main results in this paper establish new Hörmander-Mikhlin criteria for spectral and non-spectral multipliers. The key novelties which shape our approach are three. First, we control a broad class of Fourier multipliers by certain maximal operators in noncommutative $L_p$ spaces. This general principle ---exploited in Euclidean harmonic analysis during the last 40 years--- is of independent interest and might admit further applications. Second, we replace the formerly used cocycle dimensio…
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…
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 $G$.
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…
A result of Gilbert shows that every completely bounded multiplier $f$ of the Fourier algebra $A(G)$ arises from a pair of bounded continuous maps $α,β:G \rightarrow K$, where $K$ is a Hilbert space, and $f(s^{-1}t) = (β(t)|α(s))$ for all $s,t\in G$. We recast this in terms of adjointable operators acting between certain Hilbert C$^*$-modules, and show that an analogous construction works for completely bounded left multipliers of a locally compact quantum group. We find various ways to deal with right multipliers: one of these involves looking at the opposite quantum group, and this leads to …
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 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.
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…
Our main result provides a closed expression for the completely bounded Fourier multiplier norm of the spherical functions on the generalized Lorentz groups. As a corollary, we find that there is no uniform bound on the completely bounded Fourier multiplier norm of the spherical functions on the generalized Lorentz groups. We extend the latter result to the remaining connected, real rank one, simple Lie groups with finite center, and as an application we obtain that each of the above mentioned groups has a completely bounded Fourier multiplier, which is not the coefficient of a uniformly bound…
We obtain dimension free estimates for noncommutative Riesz transforms associated to conditionally negative length functions on group von Neumann algebras. This includes Poisson semigroups, beyond Bakry's results in the commutative setting. Our proof is inspired by Pisier's method and a new Khintchine inequality for crossed products. New estimates include Riesz transforms associated to fractional laplacians in $\mathbb{R}^n$ (where Meyer's conjecture fails) or to the word length of free groups. Lust-Piquard's work for discrete laplacians on LCA groups is also generalized in several ways. In th…
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.
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…
This is a survey on a notion of invariant operators, or Fourier multipliers on Hilbert spaces. This concept is defined with respect to a fixed partition of the space into a direct sum of finite dimensional subspaces. In particular this notion can be applied to the important case of $L^2(M)$ where $M$ is a compact manifold $M$ endowed with a positive measure. The partition in this case comes from the spectral properties of a a fixed elliptic operator $E$.
Let $G$ be a locally compact abelian Hausdorff topological group which is non-compact and whose Pontryagin dual $Γ$ is partially ordered. Let $Γ^{+}\subsetΓ$ be the semigroup of positive elements in $Γ$. The Hardy space $H^{2}(G)$ is the closed subspace of $L^{2}(G)$ consisting of functions whose Fourier transforms are supported on $Γ^{+}$. In this paper we consider the C*-algebra $C^{*}(\mathcal{T}(G)\cup F(C(\dot{Γ^{+}})))$ generated by Toeplitz operators with continuous symbols on $G$ which vanish at infinity and Fourier multipliers with symbols which are continuous on one point compactific…
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.
For a Banach algebra $A$ with a bounded approximate identity, we investigate the $A$-module homomorphisms of certain introverted subspaces of $A^*$, and show that all $A$-module homomorphisms of $A^*$ are normal if and only if $A$ is an ideal of $A^{**}$. We obtain some characterizations of compactness and discreteness for a locally compact quantum group $\G$. Furthermore, in the co-amenable case we prove that the multiplier algebra of $\LL$ can be identified with $\MG.$ As a consequence, we prove that $\G$ is compact if and only if $\LUC={\rm WAP}(\G)$ and $\MG\cong\mathcal{Z}({\rm LUC}(\G)^*…
Positive isometric Fourier multipliers on non-commutative $L^p$-spaces
Université Clermont Auvergne（克莱蒙奥弗涅大学）；CNRS（法国国家科学研究中心）；Laboratoire de Mathématiques de Besançon, UMR 6623, CNRS, Université Marie et Louis Pasteur（贝桑松数学实验室，UMR 6623，法国国家科学研究中心，玛丽和路易·巴斯德大学）；School of Mathematics, University of Bristol（布里斯托尔大学数学学院）
