\documentclass[a4paper,10pt]{article} % \usepackage{lmodern} \newlength{\marge} \setlength{\marge}{99pt} \addtolength{\voffset}{\marge} \addtolength{\voffset}{-125pt} \addtolength{\hoffset}{\marge} \addtolength{\hoffset}{-125pt} \setlength{\textheight}\paperheight \addtolength{\textheight}{-2\marge} \addtolength{\textheight}{-1pt} \setlength{\textwidth}\paperwidth \addtolength{\textwidth}{-2\marge} \renewcommand{\thesection}{\arabic{section}} \renewcommand{\labelenumi}{$(\alph{enumi})$} \renewcommand{\labelenumii}{\arabic{enumii}.} \renewcommand{\labelenumiii}{$(\roman{enumiii})$} % \bibliographystyle{plain} \usepackage[colorlinks=true,breaklinks=true,citecolor=blue,linkcolor=blue,urlcolor=blue,unicode]{hyperref} %\def\texorpdfstring#1#2{#1}\def\href#1#2{#2} \title{Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group} \author{Stefan Neuwirth\and \'Eric Ricard} \date{%accepted for publication in the Canadian Journal of Mathematics } \usepackage[english]{babel} \usepackage{textcase,mathdots} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amssymb,mathrsfs,amsthm} \theoremstyle{plain} \newtheorem*{Ch}{N.~G. Chebotar\"ev's formula} \newtheorem*{Chf}{Formule de N.~G. Chebotar\"ev} \newtheorem{thm}{Theorem}[section] \newtheorem{thmk}{Theorem} \newtheorem{thmf}[thm]{Th\'eor\eme} \newtheorem{lem}[thm]{Lemma} \newtheorem{lemk}[thmk]{Lemma} \newtheorem{lemf}[thm]{Lemme} \newtheorem{prp}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \newtheorem{corf}[thm]{Corollaire} \newtheorem{construction}[thm]{Random construction} \theoremstyle{definition} \newtheorem{dfn}[thm]{Definition} \newtheorem{dfnk}[thmk]{Definition} \newtheorem{dfnf}[thm]{D\'efinition} \newtheorem{pbl}[thm]{Problem} \newtheorem*{Ter}{Terminology} \newtheorem*{Not}{Notation} \newtheorem*{NotTer}{Notation and terminology} \newtheorem*{acknowledgements}{Acknowledgement} \newtheorem{???}[thm]{Conjecture} \newtheorem{ep}[thm]{Extremal problem} \newtheorem{epf}[thm]{Probl\eme extr\'emal} \newtheorem{pf}[thm]{Probl\eme} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \newtheorem{exa}[thm]{Example} \newtheorem{exe}[thm]{Exemple} \newtheorem{Q}[thm]{Question} \numberwithin{equation}{section} \def\klammer#1{(#1)} \providecommand{\abs}{\lvert#1\rvert} \providecommand{\bigabs}{\bigl\lvert#1\bigr\rvert} \providecommand{\biggabs}{\biggl\lvert#1\biggr\rvert} \providecommand{\Bigabs}{\Bigl\lvert#1\Bigr\rvert} \def\Alpha{\mathrm A} \DeclareMathOperator*{\average}{average} \def\Beta{\mathrm B} \def\block#1#2{{\setbox0=\hbox{#1\kern1ex}\leftskip=\wd0\parindent=-\wd0\par\leavevmode\box0 #2\par}} \def\borne{\mathbb B} \def\C{\mathbb C} \DeclareMathOperator{\card}{\#} \def\cb{\mathrm{cb}} \def\cnaught{\mathrm c_0} \def\col{c} \def\Col{C} \def\concat{\mathord{{}^\smallfrown}} \def\Cont{\mathrm C} \def\D{\mathbb D} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator\dist{dist} \def\doublesum{\mathop{\sum\sum}\limits} \def\e{\mkern1mu\mathrm e\mkern1mu} \def\ei#1{\mkern1mu\mathrm e\mkern2mu^{\mathrm i#1}} \def\emi#1{\mkern1mu\mathrm e\mkern2mu^{-\mathrm i#1}} \def\eps{\varepsilon} \def\fourier{\mathrm m} \def\Fourier{\mathrm M} \def\ga{\gamma} \def\Ga{\varGamma} \renewcommand{\ge}{\geqslant} \def\haar{m} \def\Hardy{\mathrm H} \def\Ht{\mathscr H} \def\iu{\mkern1mu\mathrm i\mkern1mu} \def\Id{\mathrm{Id}} \def\id{\epsilon} \def\imp{\Rightarrow} \def\Imp{\Longrightarrow} \def\io[#1,#2]{\mathopen]#1,\allowbreak#2\mathclose[} \def\iog[#1,#2]{\mathopen]#1,\allowbreak#2]} \def\bigiog[#1,#2]{\bigl]#1,\allowbreak#2\bigr]} \def\iod[#1,#2]{[#1,\allowbreak#2\mathclose[} \def\bigiod[#1,#2]{\bigl[#1,\allowbreak#2\bigr[} \def\Ell{\mathrm L} \def\la{\lambda} \def\La{\varLambda} \def\bLa{{\breve\La}} \def\HLa{{\mathaccent"707D \La}} \renewcommand{\le}{\leqslant} \def\mat{{\mathrm S}^\infty} \def\Mu{\mathrm M} \def\N{\mathbb N} \def\Nu{\mathrm N} \providecommand{\norm}{\lVert#1\rVert} \providecommand{\bignorm}{\bigl\lVert#1\bigr\rVert} \providecommand{\biggnorm}{\biggl\lVert#1\biggr\rVert} \providecommand{\Bignorm}{\Bigl\lVert#1\Bigr\rVert} \DeclareMathOperator{\pgdc}{pgdc} \def\ph{\varphi} \def\bph{{\breve\varphi}} \def\Hph{{\mathaccent"707D \ph}} \def\Prob{\mathbb P} \def\pstar{^{\scriptscriptstyle(\kern-1pt\lower0.5pt\hbox{$*$}\kern-1pt)}} \def\ppstar{^{\scriptscriptstyle(\kern-1pt*\kern-1pt)}} \def\R{\mathbb R} \def\rh{\varrho} \def\Rh{\mathrm R} \def\Rt{\mathscr T} \def\row{r} \def\Row{R} \def\Sch{\mathrm S} \def\Schur{\mathrm M} \DeclareMathOperator\sgn{sgn} \DeclareMathOperator\Hsgn{\mathaccent"707D{sgn}} \def\Sign{\mathbb S} \def\ssi{\Leftrightarrow} \def\Ssi{\Longleftrightarrow} \def\T{\mathbb T} \newcommand{\tens}{\otimes} \def\projtens{\mathop{\otimes}\limits^\wedge} \def\injtens{\mathop{\otimes}\limits^\vee} \def\th{\vartheta} \def\Th{\Theta} \DeclareMathOperator{\tr}{tr} \def\U{\mathbb U} \newcommand{\UU}{\mathscr{U}} \def\un{\chi} \def\Hun{\mathaccent"707D\un_{\Z^+}} \newcommand{\V}{\mathscr{V}} \def\Varopoulos{\mathrm V} \newcommand{\W}{\mathscr{W}} \def\vn{\Ell^\infty} \def\Walk{\mathrm W} \def\WR{\mathscr W} \def\Zeta{\mathrm Z} \def\Z{\mathbb Z} \begin{document} \maketitle \makeatletter \renewcommand{\@makefntext}{#1} \makeatother \footnotetext{Both authors were partially supported by ANR grant 06-BLAN-0015.} \begin{abstract} \noindent We inspect the relationship between relative Fourier multipliers on noncommutative Le\-besgue-Orlicz spaces of a discrete group~$\Ga$ and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum $\La\subseteq\Ga$, the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of~2, and the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than~1. \end{abstract} \section{Introduction} \noindent Let $\La$~be a subset of~$\Z$ and let $x$~be a bounded measurable function on the circle~$\T$ with Fourier spectrum in~$\La$: we write $x\in\Ell^\infty_\La$, $x\sim\sum_{k\in\La}x_kz^k$. The matrix of the associated operator $y\mapsto xy$ on~$\Ell^2$ with respect to its trigonometric basis is the Toeplitz matrix \begin{equation*} (x_{r-c})_{(r,c)\in\Z\times\Z}= \bordermatrix{% &\scriptstyle\cdots&\scriptstyle1&\scriptstyle0&\scriptstyle-1&\scriptstyle\cdots\cr \hfil\scriptstyle\vdots&\ddots&\ddots&\ddots&\ddots&\ddots\cr \hfil\scriptstyle1&\ddots&x_0&x_{1}&x_{2}&\ddots\cr \hfil\scriptstyle0&\ddots&x_{-1}&x_0&x_{1}&\ddots\cr \hfil\scriptstyle-1&\ddots&x_{-2}&x_{-1}&x_0&\ddots\cr \hfil\scriptstyle\vdots&\ddots&\ddots&\ddots&\ddots&\ddots\cr} \end{equation*} with support in $\HLa=\{(r,c):r-c\in\La\}$. This is a point of departure for the interplay of harmonic analysis and operator theory. In the general case of a discrete group~$\Ga$, the counterpart to a bounded measurable function is defined as a bounded operator on~$\ell^2_\Ga$ whose matrix has the form $(x_{rc^{-1}})_{(r,c)\in\Ga\times\Ga}$ for some sequence $(x_\ga)_{\ga\in\Ga}$. This will be the framework of the body of this article, while the introduction sticks to the case $\Ga=\Z$. We are concerned with two kinds of multipliers. A sequence $\ph=(\ph_k)_{k\in\La}$ defines \begin{itemize} \item the relative Fourier multiplication operator on trigonometric polynomials with spectrum in~$\La$ by \begin{equation} \sum_{k\in\La}x_kz^k\mapsto\sum_{k\in\La}\ph_kx_kz^k;\label{rfmo} \end{equation} \item the relative Schur multiplication operator on finite matrices with support in~$\HLa$ by \begin{equation} (x_{r,c})_{(r,c)\in\Z\times\Z}\mapsto(\Hph_{r,c}x_{r,c})_{(r,c)\in\Z\times\Z},\label{rsmo} \end{equation} where $\Hph_{r,c}=\ph_{r-c}$. \end{itemize} Marek Bo\.zejko and Gero Fendler proved that these two multipliers have the same norm. The operator~\eqref{rfmo} is nothing but the restriction of~\eqref{rsmo} to Toeplitz matrices. They noted that it is automatically \emph{completely} bounded: it has the same norm when acting on trigonometric series with operator coefficients~$x_k$, and this permits to remove this restriction. Schur multiplication is also automatically completely bounded. A part of this observation has been extended by Gilles Pisier to multipliers acting on a translation invariant Lebesgue space~$\Ell^p_\La$ and on the subspace~$\Sch^p_{\mkern-3mu\HLa}$ of elements of a Schatten-von-Neumann class supported by~$\HLa$, respectively; it yields that the complete norm of a relative Schur multiplier~\eqref{rsmo} remains bounded by the complete norm of the relative Fourier multiplier~\eqref{rfmo}. But $\Ell^p_\La$ is not a subspace of~$\Sch^p_{\mkern-3mu\HLa}$, so a relative Fourier multiplier may not be viewed anymore as the restriction of a relative Schur multiplier to Toeplitz matrices. We point out that this difficulty may be overcome by using Szeg\H{o}'s limit theorem: a bounded measurable real function on~$\T$ is the $\textrm{weak}^*$ limit of the normalised counting measure of eigenvalues of finite truncates of its Toeplitz matrix. This method also applies to Orlicz norms. \begin{thm} Let\/ $\psi\colon\R^+\to\R^+$ be a continuous nondecreasing function vanishing only at~0. The norm of the relative Fourier multiplication operator~\eqref{rfmo} on the Lebesgue-Orlicz space\/~$\Ell^\psi_\La$ is bounded by the norm of the relative Schur multiplication operator~\eqref{rsmo} on the Schatten-von-Neu\-mann-Orlicz class\/~$\Sch^\psi_{\mkern-3mu\HLa}$. \end{thm} In order to deal with complete norms, we deduce a block matrix variant of Szeg\H{o}'s limit theorem in the style of Erik B\'edos (\cite{be97}), Theorem~\ref{sz}. Note that other types of approximation are also available, as the completely positive approximation property and Reiter sequences combined with complex interpolation. They are studied in Section~\ref{sec:loc} in terms of local embeddings of~$\Ell^p$ into~$\Sch^p$. They are more canonical than Szeg\H{o}'s limit theorem, but give no access to Orlicz norms. \begin{thm} Let\/ $\psi\colon\R^+\to\R^+$ be a continuous nondecreasing function vanishing only at\/~0. The norm of the following operators is equal: \begin{itemize} \item the relative Fourier multiplication operator~\eqref{rfmo} on the Lebesgue-Orlicz space\/ $\Ell^\psi_\La(\Sch^\psi)$ of\/ $\Sch^\psi$-valued trigonometric series with spectrum in\/~$\La$; \item the relative Schur multiplication operator~\eqref{rsmo} on the Schatten-von-Neu\-mann-Orlicz class\/ $\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)$ of\/ $\Sch^\psi$-valued matrices with support in\/~$\HLa$. \end{itemize} \end{thm} See Theorems \ref{trprp}~and~\ref{trans2} for the precise statement in the general case of an amenable group~$\Ga$. An application of this theorem to the class of all unimodular Fourier multipliers yields a transfer of lacunary subsets into lacunary matrix patterns. Call~$\La$ \emph{unconditional in~$\Ell^p$} if $(z^k)_{k\in\La}$ is an unconditional basis of~$\Ell^p_\La$, and call~$\HLa$ \emph{unconditional in~$\Sch^p$} if the sequence $(\e_q)_{q\in\HLa}$ of elementary matrices is an unconditional basis of~$\Sch^p_{\mkern-3mu\HLa}$. These properties are also known as $\Lambda(p)$ if $p>2$ ($\Lambda(2)$ if $p<2$) and $\sigma(p)$, respectively; they have natural complete'' counterparts that are also known as $\Lambda(p)_\cb$ if $p>2$ ($\mathrm{K}(p)_\cb$ if $p\le2$) and $\sigma(p)_\cb$, respectively. (See Definitions \ref{unclp:def}~and~\ref{uncsp:def}). \begin{cor} Let $1\le p<\infty$. If\/ $\HLa$ is unconditional in\/~$\Sch^p$, then\/ $\La$ is unconditional in\/~$\Ell^p$. \ $\HLa$ is completely unconditional in\/~$\Sch^p$ if and only if\/ $\La$ is completely unconditional in\/~$\Ell^p$. \end{cor} See Proposition~\ref{trans:set} for the precise statement in the general case of a discrete group~$\Ga$. The two most prominent multipliers are the Riesz projection and the Hilbert transform. The first consists in letting $\ph$~be the indicator function of nonnegative integers and transfers into the upper triangular truncation of matrices. The second corresponds to the sign function and transfers into the Hilbert matrix transform. We obtain the following partial results. \begin{thm}\label{rhtrht} The norm of the matrix Riesz projection and of the matrix Hilbert transform on\/ $\Sch^\psi(\Sch^\psi)$ coincide with their norm on\/~$\Sch^\psi$. \begin{itemize} \item If\/ $p$~is a power of\/~2, then the norm of the matrix Hilbert transform on\/~$\Sch^p$ is\/ $\cot(\pi/2p)$. \item The norm of the matrix Riesz projection on\/~$\Sch^4$ is\/~$\sqrt2$. \end{itemize} \end{thm} The transfer technique lends itself naturally to the case where $\La$ contains a sumset $\Row+\Col$: if subsets $\Row'$~and~$\Col'$ are extracted so that the $r+c$ with $r\in\Row'$ and $c\in\Col'$ are pairwise distinct, they may play the role of rows and columns. Here are the consequences of the conditionality of the sequence of elementary matrices $\e_{r,c}$ in~$\Sch^p$ for $p\ne2$ and of the unboundedness of the Riesz transform on $\Sch^1$ and $\Sch^\infty$, respectively. \begin{thm}If\/ $(z^k)_{k\in\La}$ is a completely unconditional basis of\/ $\Ell^p_\La$ with\/ $p\ne2$, then\/ $\La$ does not contain sumsets\/ $\Row+\Col$ of arbitrarily large sets. If either \begin{itemize} \item the space\/ $\Ell^1_\La$ admits some completely unconditional approximating sequence, or \item the space\/ $\Cont_\La$ of continuous functions with spectrum in\/ $\La$ admits some unconditional approximating sequence, \end{itemize} then\/ $\La$ does not contain the sumset\/ $\Row+\Col$ of two infinite sets. \end{thm} The proof of the second part of this theorem consists in constructing infinite subsets $\Row'$~and~$\Col'$ and skipped block sums $\sum(T_{k_{j+1}}-T_{k_j})$ of a given approximating sequence that act like the projection on the upper triangular'' part of~$\Row'+\Col'$. See Proposition~\ref{har:lp} and Theorem~\ref{T} for the precise statement in the general case of a discrete group $\Ga$. In the case of quasi-normed Schatten-von-Neumann classes $\Sch^p$ with $p<1$, the transfer technique yields a new proof for the following result of Alexey Alexandrov and Vladimir Peller. \begin{thm} Let\/~$00$. If $\psi$~is convex, then $\Sch^\psi$~is a Banach space for the norm given by $\|x\|_{\Sch^\psi}=\inf\{a>0:\tr\psi(|x|/a)\le1\}$. Otherwise, $\Sch^\psi$ is a Fr\'echet space for the F-norm given by $\|x\|_{\Sch^\psi}=\inf\{a>0:\tr\psi(|x|/a)\le a\}$ (see \cite[Chapter~3]{or92}). This space may also be constructed as the noncommutative Lebesgue-Orlicz space~$\Ell^\psi(\tr)$ associated with a corner of the von~Neumann algebra~$\borne(\ell^2_\Col\oplus \ell^2_\Row)$ endowed with the normal, faithful, semifinite trace~$\tr$. If $\psi$~is the power function~$t\mapsto t^p$, this space is denoted~$\Sch^p$; if $p\ge1$, then $\|x\|_{\Sch^p}={(\tr|x|^p)}^{1/p}$; if $p<1$, then $\|x\|_{\Sch^p}={(\tr|x|^p)}^{1/(1+p)}$. If~$\card\Col=\card\Row=n$, then $\borne(\ell^2_\Col,\ell^2_\Row)$ identifies with the space of $n\times n$ matrices denoted~$\mat_n$, and we write~$\Sch^\psi_n$ for~$\Sch^\psi$. Let $(\Row_n\times\Col_n)$~be a sequence of finite sets such that each element of~$\Row\times\Col$ eventually is in~$\Row_n\times\Col_n$. Then the sequence of operators~$P_n\colon x\mapsto\sum_{q\in\Row_n\times \Col_n}x_q\e_q$ tends pointwise to the identity on~$\Sch^\psi$. For~$I\subseteq\Row\times\Col$, we define the space~$\Sch^\psi_I$ as the closed subspace of~$\Sch^\psi$ spanned by~$(\e_q)_{q\in I}$; this coincides with the subspace of those~$x\in \Sch^\psi$ whose support is a subset of~$I$. A \emph{relative Schur multiplier} on~$\Sch^\psi_I$ is a sequence~$\rh=(\rh_q)_{q\in I}\in\C^I$ such that the associated \emph{Schur multiplication operator}~$\Schur_\rh$ defined by~$\e_q\mapsto\rh_q\e_q$ for~$q\in I$ is bounded on~$\Sch^\psi_I$. The norm~$\|\rh\|_{\Schur(\Sch^\psi_I)}$ of~$\rh$ is defined as the norm of~$\Schur_\rh$. This norm is the supremum of the norm of its restrictions to finite rectangle sets~$\Row'\times\Col'$. We used \cite{pi98,pi01} as a reference. Let $\Ga$~be a discrete group with identity~$\id$. The \emph{reduced $\Cont^*$-algebra} of~$\Ga$ is the closed subspace spanned by the left translations~$\la_\ga$ (the linear operators defined on~$\ell^2_\Ga$ by~$\la_\ga\e_\beta=\e_{\ga\beta}$) in~$\borne(\ell^2_\Ga)$; we denote it by~$\Cont$, set in roman type. The \emph{von~Neumann algebra} of~$\Ga$ is its $\textrm{weak}^*$ closure, endowed with the normal, faithful, normalised finite trace~$\tau$ defined by~$\tau(x)=x_{\id,\id}$; we denote it by~$\vn$. Let $\psi\colon\R^+\to\R^+$ be a continuous nondecreasing function vanishing only at~0. We define the noncommutative Lebesgue-Orlicz space~$\Ell^\psi$ of $\Ga$ as the completion of $\vn$ with respect to the norm given by $\|x\|_{\Ell^\psi}=\inf\{a>0:\tau(\psi(|x|/a))\le1\}$ if $\psi$~is convex, and with respect to the F-norm given by $\|x\|_{\Ell^\psi}=\inf\{a>0:\tau(\psi(|x|/a))\le a\}$ otherwise. If $\psi$~is the power function~$t\mapsto t^p$, this space is denoted~$\Ell^p$; if $p\ge1$, then $\|x\|_{\Ell^p}={\tau(|x|^p)}^{1/p}$; if $p<1$, then $\|x\|_{\Ell^p}={\tau(|x|^p)}^{1/(1+p)}$. The \emph{Fourier coefficient} of~$x$ at~$\ga$ is~$x_\ga=\tau(\la_\ga^*x)=x_{\ga,\id}$ and its \emph{Fourier series} is~$\sum_{\ga\in\Ga}x_\ga\la_\ga$. The \emph{spectrum} of an element~$x$ is~$\{\ga\in\Ga:x_\ga\ne0\}$. Let $X$~be the $\Cont^*$-algebra~$\Cont$ or the space~$\Ell^\psi$ and let~$\La\subseteq\Ga$; then we define $X_\La$~as the closed subspace of~$X$ spanned by the~$\la_\ga$ with~$\ga\in\La$. We skip the general question of when this coincides with the subspace of those~$x\in X$ whose spectrum is a subset of~$\La$, but note that this is the case if $\Ga$ is an amenable group (or if $\Ga$ has the AP and $\vn$~has the QWEP by \cite[Theorem~4.4]{jr03}) and $\psi$~is the power function~$t\mapsto t^p$. Note also that our definition of $X_\La$ makes it a subspace of the \emph{heart} of $X$: if $x\in X_\La$, then $\tau(\psi(|x|/a))$ is finite for all $a>0$. A \emph{relative Fourier multiplier} on~$X_\La$ is a sequence~$\ph=(\ph_\ga)_{\ga\in\La}\in\C^\La$ such that the associated Fourier multiplication operator~$\Fourier_\ph$ defined by~$\la_\ga\mapsto\ph_\ga\la_\ga$ for~$\ga\in\La$ is bounded on~$X_\La$. The norm~$\|\ph\|_{\Fourier(X_\La)}$ of~$\ph$ is defined as the norm of~$\Fourier_\ph$. Fourier multipliers on the whole of the $\Cont^*$-algebra $\Cont$ are also called \emph{multipliers of the Fourier algebra ${\mathrm A}(\Ga)$} (which may be identified with $\Ell^1$); they form the set $\Fourier(\mathrm{A}(\Ga))$. The space~$\Sch^\psi(\Sch^\psi)$ is the space of those compact operators~$x$ from~$\ell^2\otimes\ell^2_\Col$ to~$\ell^2\otimes\ell^2_\Row$ such that $\|x\|_{\Sch^\psi(\Sch^\psi)}=\inf\{a:\tr\otimes\tr\psi(|x|/a)\le1\}$: it is the noncommutative Lebesgue-Orlicz space~$\Ell^\psi(\tr\otimes\tr)$ associated with a corner of the von~Neumann algebra~$\borne(\ell^2)\otimes\borne(\ell^2_\Col\oplus \ell^2_R)$. One may think of~$\Sch^\psi(\Sch^\psi)$ as the $\Sch^\psi$-valued Schatten-von-Neumann class; we define the matrix coefficient of~$x$ at~$q$ by~$x_q = (\Id_{\Sch^\psi}\otimes\tr)\allowbreak \bigl((\Id_{\ell^2}\otimes\e_q^*)x\bigr)\in\Sch^\psi$ and its matrix representation by $\sum_{q\in\Row\times\Col}x_q\tens\e_q$. The support of~$x$ and the subspace~$\Sch^\psi_I(\Sch^\psi)$ are defined in the same way as~$\Sch^\psi_I$. Similarly, the space~$\Ell^\psi(\tr\otimes\tau)$ is the noncommutative Lebesgue-Orlicz space associated with the von~Neumann algebra~$\borne(\ell^2)\otimes\vn=\Ell^\infty(\tr\otimes\tau)$. One may think of~$\Ell^\psi(\tr\otimes\tau)$ as the $\Sch^\psi$-valued noncommutative Lebesgue space; we define the Fourier coefficient of~$x$ at~$\ga$ by~$x_\ga=(\Id_{\Sch^\psi}\otimes\tau)\bigl((\Id_{\ell^2}\otimes\la_\ga^*)x\bigr)\in\Sch^\psi$ and its Fourier series by~$\sum_{\ga\in\Ga}x_\ga\tens\la_\ga$; the spectrum of~$x$ is defined accordingly. The subspace~$\Ell^\psi_\La(\tr\otimes\tau)$ is the closed subspace of~$\Ell^\psi(\tr\otimes\tau)$ spanned by the~$x\otimes\la_\ga$ with~$x\in\Sch^\psi$ and~$\ga\in\La$. An operator~$T$ on~$\Sch^\psi_I$ is \emph{bounded on $\Sch^\psi_I(\Sch^\psi)$} if the linear operator $\Id_{\Sch^\psi}\otimes T$ defined by~$x\tens y\mapsto x\tens T(y)$ for~$x\in \Sch^\psi$ and $y$ in~$\Sch^\psi_I$ on finite tensors extends to a bounded operator~$\Id_{\Sch^\psi}\otimes T$ on~$\Sch^\psi_I(\Sch^\psi)$. The norm of a Schur multiplier~$\rh$ on $\Sch^\psi_I(\Sch^\psi)$ is defined as the norm of~$\Id_{\Sch^\psi}\otimes\Schur_\rh$. Similar definitions hold for an operator~$T$ on~$\Ell^\psi_\La$; the norm of a Fourier multiplier~$\ph$ on~$\Ell^\psi_\La(\tr\otimes\tau)$ is the norm of $\Id_\Sch^\psi\otimes\Fourier_\ph$ on~$\Ell^\psi_\La(\tr\otimes\tau)$. Let $\psi$~be the power function~$t\mapsto t^p$ with $p\ge1$; the norms on $\Sch^p(\Sch^p)$ and $\Ell^p(\tr\otimes\tau)$ describe the canonical operator space structure on $\Sch^p$~and~$\Ell^p$, respectively (see \cite[Corollary~1.4]{pi98}); we should rather use the notation $\Sch^p[\Sch^p]$ and $\Sch^p[\Ell^p]$. This explains the following terminology. An operator~$T$ on~$\Sch^p_I$ is \emph{completely bounded} (c.b.) if $\Id_{\Sch^p}\otimes T$~is bounded on~$\Sch^p_I(\Sch^p)$; the norm of~$\Id_{\Sch^p}\otimes T$ is the \emph{complete norm} of~$T$ (compare \cite[Lemma~1.7]{pi98}). The complete norm~$\|\rh\|_{\Schur_\cb(\Sch^p_I)}$ of a Schur multiplier~$\rh$ is defined as the complete norm of~$\Schur_\rh$. Note that the complete norm of a Schur multiplier~$\rh$ on~$\Sch^\infty_I$ is equal to its norm (\cite[Theorem~3.2]{pps89}): \ $\|\rh\|_{\Schur_\cb(\Sch^\infty_I)}=\|\rh\|_{\Schur(\Sch^\infty_I)}$. The complete norm~$\|\ph\|_{\Fourier_\cb(\Ell^p_\La)}$ of a Fourier multiplier~$\ph$ is defined as the complete norm of~$\Fourier_\ph$. The complete norm of an operator~$T$ on~$\Cont_\La$ is the norm of~$\Id_{\Sch^\infty}\otimes T$ on the subspace of~$\Sch^\infty\otimes\Cont$ spanned by the~$x\otimes\la_\ga$ with~$x\in\Sch^\infty$ and~$\ga\in\La$. In the case $\La=\Ga$, $\ph$ is also called a \emph{c.b.\ multiplier of the Fourier algebra ${\mathrm A}(\Ga)$} and one writes $\ph\in\Fourier_{\cb}(\mathrm A(\Ga))$. If $\Ga$~is amenable, the complete norm of a Fourier multiplier~$\ph$ on~$\Cont_\La$ is equal to its norm: \ $\|\ph\|_{\Fourier_\cb(\Cont_\La)}=\|\ph\|_{\Fourier(\Cont_\La)}$ (this follows from \cite[Corollary~1.8]{dh85} as shown by the proof of Theorem~\ref{trans2}$\,(c)$). An element whose norm is at most~1 is \emph{contractive}, and if its complete norm is at most~1, it is \emph{completely} contractive. If $\Ga$~is abelian, let $G$~be its dual group and endow it with its unique normalised Haar measure~$m$. Then the Fourier transform identifies the $\Cont^*$-algebra~$\Cont$ as the space of continuous functions on~$G$, \ $\Ell^\infty$ as the space of classes of bounded measurable functions on~$(G,m)$, \ $\Ell^\psi$ as the Lebesgue-Orlicz space of classes of~$\psi$-integrable functions on~$(G,m)$, \ $\tau(x)$ as~$\int_Gx(g)\,\mathrm dm(g)$, \ $\Ell^\psi(\tr\otimes\tau)$ as the $\Sch^\psi$-valued Lebesgue-Orlicz space $\Ell^\psi(\Sch^\psi)$ and $x_\ga$ as~$\hat x(\ga)$. \end{NotTer} \section{Transfer between Fourier and Schur multipliers}\label{sec:tr} Let $\La$~be a subset of a discrete group~$\Ga$ and let $\ph$~be a relative Fourier multiplier on~$\Cont_\La$, the closed subspace spanned by $(\la_\ga)_{\ga\in\La}$ in the reduced $\Cont^*$-algebra of $\Ga$. Let $x\in\Cont_\La$; the matrix of~$x$ is \emph{constant down the diagonals} in the sense that for every~$(r,c)\in\Ga\times\Ga$, \ $x_{r,c}=x_{rc^{-1},\id}=x_{rc^{-1}}$. We say that $x$~is a \emph{Toeplitz} operator on~$\ell^2_\Ga$. Furthermore, the matrix of the Fourier product~$\Fourier_\ph x$ of~$\ph$ with~$x$ is given by~$(\Fourier_\ph x)_{r,c}=\ph_{rc^{-1}}x_{r,c}$. This equality shows that if we set~$\HLa=\{(r,c)\in\Ga\times\Ga:\row\col^{-1}\in \La\}$~and~$\Hph_{r,c}=\ph_{\row\col^{-1}}$, then $\Fourier_\ph x$~is the Schur product $\Schur_\Hph x$ of~$\Hph$ with~$x$. We have transferred the Fourier multiplier~$\ph$ into the Schur multiplier~$\Hph$. This proves at once that the norm of the Fourier multiplier~$\ph$ on~$\Cont_\La$ is the norm of the Schur multiplier~$\Hph$ on the subspace of Toeplitz elements of~$\borne(\ell^2_\Ga)$ with support in~$\HLa$, and that the same holds for complete norms. We shall now provide the means to generalise this identification to the setting of Lebesgue-Orlicz spaces $\Ell^\psi$. We shall bypass the main obstacle, that $\Ell^\psi$~may not be considered as a subspace of~$\Sch^\psi$, by the Szeg\H{o} limit theorem as stated by Erik B\'edos (\cite{be97}). Consider a discrete amenable group~$\Ga$; it admits a \emph{F{\o}lner averaging net of sets} $(\Ga_\iota)$, that is, \begin{itemize} \item each~$\Ga_\iota$ is a finite subset of~$\Ga$; \item $\card(\ga\Ga_\iota\Delta\Ga_\iota)=o(\card\Ga_\iota)$ for each~$\ga\in\Ga$. \end{itemize} Each set~$\Ga_\iota$ corresponds to the orthogonal projection~$p_\iota$ of~$\ell^2_\Ga$ onto its $(\card\Ga_\iota)$-di\-men\-sio\-nal subspace of sequences supported by~$\Ga_\iota$. The \emph{truncate} of a selfadjoint operator~$y\in\borne(\ell^2_\Ga)$ with respect to~$\Ga_\iota$ is $y_\iota=p_\iota^{\vphantom{*}} yp_\iota^*$; it has $\card\Ga_\iota$ eigenvalues~$\alpha_j$, counted with multiplicities, and its \emph{normalised counting measure of eigenvalues} is \begin{equation*} \mu_\iota=\frac1{\card\Ga_\iota}\sum_{j=1}^{\card\Ga_\iota}\delta_{\alpha_j}. \end{equation*} If $y$~is a Toeplitz operator, that is, if $y\in\vn$, Erik B\'edos (\cite[Theorem~10]{be97}) proved that $(\mu_\iota)$ converges $\textrm{weak}^*$ to the \emph{spectral measure} of~$y$ with respect to~$\tau$, which is the unique Borel probability measure~$\mu$ on~$\R$ such that \begin{equation*} \tau(f(y))=\int_\R f(\alpha)\mathrm d\mu(\alpha) \end{equation*} for every continuous function~$f$ on~$\R$ that tends to zero at infinity. If $\Ga$~is abelian, then $y$~may be identified as the class of a real-valued bounded measurable function on the group~$G$ dual to~$\Ga$ and $\mu$~is the distribution of~$y$. Let us now state and prove the $\Ell^\psi$ version of the identification described at the beginning of this section. \begin{thm} \label{trprp} Let\/ $\Ga$~be a discrete amenable group and let\/ $\psi\colon\R^+\to\R^+$ be a continuous nondecreasing function vanishing only at~0. Let\/ $\La\subseteq\Ga$ and\/ $\ph\in\C^\La$. Consider the associated Toeplitz set\/ $\HLa=\{(r,c)\in\Ga\times\Ga:\row\col^{-1}\in\La\}$ and the Toeplitz matrix defined by\/ $\Hph_{r,c}=\ph_{\row\col^{-1}}$. The norm of the relative Fourier multiplier\/~$\ph$ on\/~$\Ell^\psi_\La$ is bounded by the norm of the relative Schur multiplier\/~$\Hph$ on\/~$\Sch^\psi_{\mkern-3mu\HLa}$. \end{thm} \begin{proof} A Toeplitz matrix has the form $(x_{rc^{-1}})_{(r,c)\in \HLa}$. Our definition of the space~$\Ell^\psi_\La$ (in the section on Notation and terminology) ensures that we may suppose that only a finite number of the~$x_\ga$ are nonzero for the computation of the norm of~$\ph$. Then $(x_{rc^{-1}})_{(r,c)\in \HLa}$ is the matrix of the operator $x=\sum_{\ga\in \La}x_\ga\la_\ga$ for the canonical basis of~$\ell^2_\Ga$. Let~$y=x^*x$ and let $\tilde\psi$~be a continuous function with compact support such that $\tilde\psi(t)=\psi(t)$ on $[0,\|y\|]$. By Szeg\H{o}'s limit theorem, \begin{equation*} \frac1{\#\Ga_\iota}\tr\psi(y_\iota)= \frac1{\#\Ga_\iota}\tr\tilde\psi(y_\iota) \to\tau(\tilde\psi(y))=\tau(\psi(y)). \end{equation*} We have $y_\iota={(xp_\iota^*)}^*(xp_\iota^*)$; let us describe how $\Hph$ acts on~$xp_\iota^*$. Schur multiplication with~$\Hph$ transforms the matrix of~$xp_\iota^*$, that is, the truncated Toeplitz matrix~$(x_{rc^{-1}})_{(r,c)\in \HLa\cap\Ga\times\Ga_\iota}$, into the matrix~$(\ph_{rc^{-1}}x_{rc^{-1}})_{(r,c)\in \HLa\cap\Ga\times\Ga_\iota}$ so that it transforms~$xp_\iota^*$ into~$(\Fourier_\ph x)p_\iota^*$. \end{proof} \begin{rem} In the case of a finite abelian group, no limit theorem is needed. This case was considered in~\cite[Proposition~2.5$\,(b)$]{ne06}; compare with \cite[Chapter 6, Lemma 3.8]{pe03}. \end{rem} \begin{rem}\label{trrem} Our technique proves in fact that the norm of a Fourier multiplier is the upper limit of the norm of the corresponding relative Schur multipliers on subspaces of truncated Toeplitz matrices. We ignore whether or not it is actually their supremum. \end{rem} Remark~\ref{b-cb} illustrates that the two norms in Theorem~\ref{trprp} are different in general. This is not so in the $\Sch^\psi$-valued case because of the following argument. It has been used (first in~\cite{bf84}, see~\cite[Proposition~D.6]{bo08}) to show that the complete norm of the Fourier multiplier~$\ph$ on~$\Cont_\La$ bounds the complete norm of the Schur multiplier~$\Hph$ on $\Sch^\infty_{\mkern-3mu\HLa}$, so that we have in full generality $\|\ph\|_{\Fourier_\cb(\Cont_\La)}=\|\Hph\|_{\Schur_\cb(\Sch^\infty_{\mkern-3mu\HLa})}$. \begin{lem} \label{trans1} Let\/ $\Ga$~be a discrete group and let\/ $\Row$~and\/~$\Col$ be subsets of\/~$\Ga$. With\/~$\La\subseteq\Ga$ associate\/~$\HLa=\{(r,c)\in\Row\times\Col:\row\col^{-1}\in \La\}$; given\/~$\ph\in\C^\La$, define\/~$\Hph\in\C^\HLa$ by\/~$\Hph_{r,c}=\ph_{\row\col^{-1}}$. Let\/ $\psi\colon\R^+\to\R^+$ be a continuous nondecreasing function vanishing only at\/~0. The norm of the relative Schur multiplier\/~${\Hph}$ on\/~$\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)$ is bounded by the norm of the relative Fourier multiplier\/~$\ph$ on\/~$\Ell^\psi_\La(\tr\otimes\tau)$. \end{lem} \begin{proof} We adapt the argument in~\cite[Lemma~8.1.4]{pi98}. Let~$x_q\in \Sch^\psi$, of which only a finite number are nonzero. The space~$\Ell^\psi(\tr\otimes\tr\otimes\tau)$ is a left and right $\Ell^\infty(\tr\otimes\tr\otimes\tau)$-module, and~$\sum_{\ga\in\Ga}\e_{\ga\ga}\tens\la_\ga$ is a unitary in~$\Ell^\infty(\tr\otimes\tau)$ so that \begin{multline*} \Bigl\|\sum_{q\in \HLa}x_q\tens\e_q\Bigr\|_{\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)}\\ \begin{aligned} &=\Bigl\|\Bigl(\Id\otimes\sum_{r\in\Row}\e_{r,r}\tens\la_\row\Bigr) \Bigl(\sum_{q\in \HLa}x_q\tens\e_q\tens\la_\id\Bigr) \Bigl(\Id\otimes\sum_{c\in\Col}\e_{c,c}\tens\la_\col^*\Bigr)\Bigr\|_{\Ell^\psi(\tr\otimes\tr\otimes\tau)}\\ &= \biggl\| \sum_{(r,c)\in \HLa} x_{r,c}\tens\e_{r,c}\tens\la_{\row\col^{-1}}\biggr\|_{\Ell^\psi(\tr\otimes\tr\otimes\tau)}\\ &= \biggl\| \sum_{\ga\in \La} \biggl(\sum_{\row\col^{-1}=\ga} x_{r,c}\tens\e_{r,c}\biggr)\tens\la_\ga\biggr\|_{\Ell^\psi_\La(\tr\otimes\tr\otimes\tau)}. \end{aligned} \end{multline*} This yields an isometric embedding of~$\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)$ in~$\Ell^\psi_\La(\tr\otimes\tr\otimes\tau)$. As $\Sch^\psi(\Sch^\psi)$~is the Schatten-von-Neumann-Orlicz class for the Hilbert space~$\ell^2\otimes\ell^2_\Ga$, which may be identified with~$\ell^2$, \begin{multline*} \Bigl\| \sum_{q\in \HLa} x_q\tens\Hph_q\e_q \Bigr\|_{\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)} = \biggl\| \sum_{\ga\in \La} \biggl( \sum_{\row\col^{-1}=\ga} x_{r,c}\tens\e_{r,c} \biggr)\tens\ph_\ga \la_\ga \biggr\|_{\Ell^\psi_\La(\tr\otimes\tr\otimes\tau)}\\ \le \|\Id_{\Sch^\psi}\otimes\Fourier_\ph\| \Bigl\| \sum_{q\in \HLa} x_q\tens\e_q \Bigr\|_{\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)}.\tag*{\qedhere} \end{multline*} \end{proof} \begin{rem} \label{transg} This proof also shows the following transfer: let $(r_i)$ and $(c_j)$ be sequences in~$\Ga$, consider~$\bLa=\{(i,j)\in\N\times\N:r_ic_j\in\La\}$ and define~$\bph\in\C^\bLa$ by~$\bph(i,j)=\ph(r_ic_j)$. Then the norm of the relative Schur multiplier~${\bph}$ on~$\Sch^\psi_{\bLa}(\Sch^\psi)$ is bounded by the norm of the relative Fourier multiplier~$\Id_{\Sch^\psi}\otimes\Fourier_\ph$ on~$\Ell^\psi_\La(\tr\otimes\tau)$ (compare with \cite[Theorem~6.4]{pi01}). In particular, if the $r_ic_j$ are pairwise distinct, this permits us to transfer every Schur multiplier, not just the Toeplitz ones. See \cite[Section~11]{ne06} for applications of this transfer. \end{rem} We shall now prove that the two norms in this lemma are in fact equal. As we want to compute norms of multipliers on $\Sch^\psi$-valued spaces, we shall generalise the Szeg\H{o} limit theorem to the block matrix case, which was not considered in \cite{be97}. This is the analogue of the scalar case for selfadjoint elements $y\in\mat_n\otimes\vn$, whose \emph{$\mat_n$-valued spectral measure}~$\mu$ is defined by \begin{equation*} \int_\R f(\alpha)\mathrm d\mu(\alpha)=\Id_{\mat_n}\otimes\tau(f(y)) \end{equation*} for every continuous function $f$ on $\R$ that tends to zero at infinity. The orthogonal projection~$\tilde p_\iota=\Id_{\ell^2_n}\otimes p_\iota$ defines the truncate~$y_\iota=\tilde p_\iota^{\vphantom{*}} y\tilde p_\iota^*\in\mat_n\otimes\borne(\ell^2_{\Ga_\iota})$, and the \emph{$\mat_n$-valued normalised counting measure of eigenvalues}~$\mu_\iota$ by \begin{equation*} \int_\R f(\alpha)\mathrm d\mu_\iota(\alpha)=\Id_{\mat_n}\otimes\frac{\tr}{\card\Ga_\iota}(f(y_\iota)) \end{equation*} for every continuous function $f$ on $\R$ that tends to zero at infinity. \begin{thm}[Matrix Szeg\H{o} limit theorem] \label{sz} Let\/ $\Ga$~be a discrete amenable group and let\/ $(\Ga_\iota)$~be a F{\o}lner averaging net for\/~$\Ga$. Let\/ $y$~be a selfadjoint element of\/~$\mat_n\otimes\vn$. The net\/~$(\mu_\iota)$ of\/~$\mat_n$-valued normalised counting measures of eigenvalues of the truncates of\/~$y$ with respect to\/~$\Ga_\iota$ converges in the\/ $\textrm{weak}^*$ topology to the spectral measure of\/~$y$: \begin{equation*} \int_\R f(\alpha)\mathrm d\mu_\iota(\alpha)\to\Id_{\mat_n}\otimes\tau(f(y)) \end{equation*} for every continuous function\/~$f$ on\/~$\R$ that tends to zero at infinity. \end{thm} \begin{proof}[Sketch of proof] We first suppose that $y=\sum_{\ga\in\Ga}y_\ga\tens\la_\ga$ with only a finite number of the~$y_\ga\in\mat_n$ nonzero. The $\mat_n$-valued matrix of the truncate~$y_\iota$ of~$y$ for the canonical basis of~$\ell^2_{\Ga_\iota}$ is~$(y_{rc^{-1}})_{(r,c)\in\Ga_\iota\times\Ga_\iota}$. As the truncates~$y_\iota$ of~$y$ are uniformly bounded, it suffices to prove that \begin{equation*} \Id\otimes\frac\tr{\#\Ga_\iota}(y_\iota^k)\to\Id\otimes\tau(y^k) \end{equation*} for every~$k$. This is trivial if~$k=0$. If~$k=1$, then \begin{equation*} \Id\otimes\frac\tr{\#\Ga_\iota}(y_\iota) =\frac1{\#\Ga_\iota}\sum_{\col\in\Ga_\iota}y_{\col,\col} =\Id\otimes\tau(y) \end{equation*} as~$y_{\col,\col}=y_{\col\col^{-1}}=y_\id$. If~$k\ge2$, the same formula holds with~$y^k$ instead of~$y$: \begin{equation*} \Id\otimes\tau(y^k)=\Id\otimes\frac\tr{\#\Ga_\iota} (\tilde p_\iota^{\vphantom{*}} y^k\tilde p_\iota^*), \end{equation*} so that we wish to prove \begin{equation*} \Id\otimes\tr(\tilde p_\iota^{\vphantom{*}} y^k\tilde p_\iota^* -{(\tilde p_\iota^{\vphantom{*}} y\tilde p_\iota^*)}^k) =o(\card\Ga_\iota). \end{equation*} Note that \begin{equation*} \bignorm{\Id\otimes\tr\bigl(\tilde p_\iota^{\vphantom{*}} y^k\tilde p_\iota^*-{(\tilde p_\iota^{\vphantom{*}} y\tilde p_\iota^*)}^k\bigr)}_{\Sch^1_n} \le\|\tilde p_\iota^{\vphantom{*}} y^k\tilde p_\iota^*-{(\tilde p_\iota^{\vphantom{*}} y\tilde p_\iota^*)}^k\|_{\Sch^1(\Sch^1_n)}. \end{equation*} Lemma~5 in \cite{be97} provides the following estimate. As \begin{equation*} \tilde p_\iota^{\vphantom{*}} y^k\tilde p_\iota^*-{(\tilde p_\iota^{\vphantom{*}} y\tilde p_\iota^*)}^k =\tilde p_\iota^{\vphantom{*}} y^{k-1}(y\tilde p_\iota^*-\tilde p_\iota^*\tilde p_\iota^{\vphantom{*}} y\tilde p_\iota^*) +(\tilde p_\iota^{\vphantom{*}} y^{k-1}\tilde p_\iota^*-{(\tilde p_\iota^{\vphantom{*}} y\tilde p_\iota^*)}^{k-1})\tilde p_\iota^{\vphantom{*}}y\tilde p_\iota^*, \end{equation*} an induction yields \begin{equation*} \|\tilde p_\iota^{\vphantom{*}} y^k\tilde p_\iota^*-{(\tilde p_\iota^{\vphantom{*}} y\tilde p_\iota^*)}^k\|_{\Sch^1(\Sch^1_n)} \le(k-1)\|y\|_{\mat_n\otimes\vn}^{k-1}\|y\tilde p_\iota^*-\tilde p_\iota^*\tilde p_\iota^{\vphantom{*}} y\tilde p_\iota^*\|_{\Sch^1(\Sch^1_n)}. \end{equation*} It suffices to consider the very last norm for each term~$y_\ga\tens\la_\ga$ of~$y$: let~$h\in\ell^2_n$ and~$\beta\in\Ga$; as \begin{equation*} \bigl((y_\ga\tens\la_\ga)\tilde p_\iota^*-\tilde p_\iota^*\tilde p_\iota^{\vphantom{*}}(y_\ga\tens\la_\ga)\tilde p_\iota^*\bigr)(h\otimes\e_\beta) = \begin{cases} y_\ga(h)\e_{\ga\beta}&\text{if $\beta\in\Ga_\iota$ and $\ga\beta\notin\Ga_\iota$}\\ 0&\text{otherwise,} \end{cases} \end{equation*} the definition of a F{\o}lner averaging net yields \begin{equation*} \|(y_\ga\tens\la_\ga)\tilde p_\iota^* -\tilde p_\iota^*\tilde p_\iota^{\vphantom{*}} (y_\ga\tens\la_\ga)\tilde p_\iota^*\| _{\Sch^1(\Sch^1_n)} \le\#(\Ga_\iota\setminus\ga^{-1}\Ga_\iota)\|y_\ga\|_{\Sch^1_n} =o(\card\Ga_\iota). \end{equation*} An approximation argument as in the proof of \cite[Proposition~4]{be97} permits us to conclude for~$y\in\mat_n\otimes\vn$. \end{proof} Here is the promised strengthening of Lemma~\ref{trans1} together with three variants. \begin{thm}\label{trans2} Let\/ $\Ga$~be a discrete amenable group. Let\/ $\La\subseteq\Ga$ and\/ $\ph\in\C^\La$. Consider the associated Toeplitz set\/~$\HLa=\{(r,c)\in\Ga\times\Ga:\row\col^{-1}\in \La\}$ and the Toeplitz matrix defined by\/~$\Hph_{r,c}=\ph_{\row\col^{-1}}$. \begin{enumerate} \item Let\/ $\psi\colon\R^+\to\R^+$ be a continuous nondecreasing function vanishing only at\/~0. The norm of the relative Fourier multiplier\/~$\ph$ on\/~$\Ell^\psi_\La(\tr\otimes\tau)$ and the norm of the relative Schur multiplier\/~${\Hph}$ on\/~$\Sch^\psi_{\mkern-3mu\HLa}(\Sch^\psi)$ are equal. \item Let\/ $p\ge1$. The complete norm of the relative Fourier multiplier\/~$\ph$ on\/~$\Ell^p_\La$ and the complete norm of the relative Schur multiplier\/~$\Hph$ on\/~$\Sch^p_{\mkern-3mu\HLa}$ are equal: \begin{equation*} \|\ph\|_{\Fourier_\cb(\Ell^p_\La)}=\|\Hph\|_{\Schur_\cb(\Sch^p_{\mkern-3mu\HLa})}. \end{equation*} \item The norm of the relative Fourier multiplier\/~$\ph$ on\/~$\Cont_\La$, its complete norm, the norm of the relative Schur multiplier\/~$\Hph$ on\/~$\Sch^\infty_{\mkern-3mu\HLa}$, and its complete norm are equal: \begin{equation*} \|\ph\|_{\Fourier(\Cont_\La)} =\|\ph\|_{\Fourier_\cb(\Cont_\La)} =\|\Hph\|_{\Schur_\cb(\Sch^\infty_{\mkern-3mu\HLa})} =\|\Hph\|_{\Schur(\Sch^\infty_{\mkern-3mu\HLa})}. \end{equation*} \item Suppose that\/ $\La=\Ga$. The norm of the Fourier algebra multiplier\/~$\ph$, its complete norm, the norm of the Schur multiplier\/~$\Hph$ on\/~$\Sch^\infty$, and its complete norm are equal: \begin{equation*} \|\ph\|_{\Fourier(\mathrm{A}(\Ga))} =\|\ph\|_{\Fourier_\cb(\mathrm A(\Ga))} =\|\Hph\|_{\Schur_\cb(\Sch^\infty)} =\|\Hph\|_{\Schur(\Sch^\infty)}. \end{equation*} \end{enumerate} \end{thm} \begin{proof} $(a)$. Combine the argument in Theorem~\ref{trprp} with the matrix Szeg\H{o} limit theorem and apply Lemma~\ref{trans1}. $(c)$. Recall that the complete norm of a Schur multiplier~$\Hph$ on~$\Sch^\infty_{\mkern-3mu\HLa}$ is equal to its norm (\cite[Theorem~3.2]{pps89}). Recall also that the norm of a Fourier multiplier~$\chi$ on~$\Cont$ is equal to its complete norm, because $\Ga$~is amenable. Moreover, it coincides with the norm of $\chi$ in $\mathrm{A}(\Ga)$ (\cite[Corollary~1.8]{dh85}). Let $\ph$ be a relative contractive Fourier multiplier on $\Cont_\La$; compose it with the trivial character of $\Ga$ to obtain a contractive form on $\Cont_\La$. Then, by the Hahn-Banach extension theorem, $\ph$ is the restriction of a contractive element $\chi$ in $\mathrm{A}(\Ga)$. Now $\chi$ is a completely contractive Fourier multiplier on $\Cont$, and so is $\ph$ on $\Cont_\La$. The conclusion follows from $(a)$ and $(b)$. \end{proof} \section{\texorpdfstring{Local embeddings of~$\Ell^p$ into~$\Sch^p$}{Local embeddings of Lp into Sp}}\label{sec:loc} \noindent The proof of Theorem~\ref{trprp} can be interpreted as an embedding of $\Ell^\psi$ into an ultraproduct of finite-dimensional spaces $\Sch^\psi_n$ that intertwines Fourier and Toeplitz Schur multipliers. If we restrict ourselves to power functions $\psi\colon t\mapsto t^p$ with $p\ge 1$, such embeddings are well known and the proof of Theorem~\ref{trans2} does not need the full strength of the matrix Szeg\H{o} limit theorem but only the existence of such embeddings. In this section, we explain two ways to obtain them by interpolation. The first way is to extend the classical result that the reduced $\Cont^*$-algebra~$\Cont$ of a discrete group $\Ga$ has the completely positive approximation property if $\Ga$ is amenable. We follow the approach of \cite[Theorem~2.6.8]{bo08}. Let $\Ga$ be a discrete amenable group and let $\Ga_\iota$ be a F{\o}lner averaging net of sets. As above, we denote by $p_\iota$ the orthogonal projection from $\ell^2_\Ga$ to $\ell^2_{\Ga_\iota}$. Define the compression~$\phi_\iota$ and the embedding $\psi_\iota$ by \begin{equation} \label{phipsi} \begin{aligned}[t] \phi_\iota\colon\Cont&\to\borne(\ell^2_{\Ga_\iota})\\ x&\mapsto p_\iota x p_\iota^* \end{aligned} \quad\text{and}\quad \begin{aligned}[t] \psi_\iota\colon\borne(\ell^2_{\Ga_\iota})&\to\Cont\\ \e_{r,c}&\mapsto(1/\card \Ga_\iota)\la_r\la_{c^{-1}}. \end{aligned} \end{equation} If we endow $\borne(\ell^2_{\Ga_\iota})$ with the normalised trace, these maps are unital completely positive, trace preserving (and normal), and the net $(\psi_\iota\phi_\iota)$ converges pointwise to the identity of $\Cont$. One can therefore extend them by interpolation to completely positive contractions on the respective noncommutative Lebesgue spaces. Recall that $\Ell^p(\borne(\ell^2_{\Ga_\iota}), (1/\card\Ga_\iota)\tr)$ is ${(\card\Ga_\iota)}^{-1/p}\Sch^p_{\card \Ga_\iota}$. We get a net of complete contractions \begin{equation*} \tilde \phi_\iota \colon \Ell^p\to {(\card\Ga_\iota)}^{-1/p}\Sch^p_{\card \Ga_\iota} \quad\text{and}\quad \tilde \psi_\iota\colon {(\card\Ga_\iota)}^{-1/p}\Sch^p_{\card \Ga_\iota} \to \Ell^p \end{equation*} such that $(\tilde\psi_\iota\tilde\phi_\iota)$ converges pointwise to the identity of $\Ell^p$. Moreover, the definitions~\eqref{phipsi} show that these maps also intertwine Fourier and Toeplitz Schur multipliers. This approach is more canonical, as it allows us to extend the transfer to vector-valued spaces in the sense of \cite[Chapter 3]{pi98}. Recall that for any hyperfinite semifinite von Neumann algebra $M$ and any operator space $E$, one can define $\Ell^p(M,E)$. For $p=\infty$, this space is defined as $M\tens_{\min}E$; for $p=1$, this space is defined as $M_*^{\mathrm{op}}\hat\tens E$; these spaces form an interpolation scale for the complex method when $1\le p\le \infty$. For us, $M$ will be $\borne(\ell^2)$ or the group von Neumann algebra $\vn$. As the maps $\psi_\iota$ and $\phi_\iota$ are unital completely positive and trace preserving and normal, they define simultaneously complete contractions on $M$ and $M_*$. By interpolation, the maps $\psi_\iota\tens \Id_E$ and $\phi_\iota\tens \Id_E$ are still complete contractions on the spaces $\Ell_p(E)$ and $\Sch^p[E]$. Let $\ph\in\C^\Ga$; the transfer shows that the norm of $\Id_E\otimes\Fourier_\ph$ on~$\Ell^p(E)$ is bounded by the norm of $\Id_E\otimes\Schur_\Hph$ on~$\Sch^p[E]$ and that their complete norms coincide. In formulas, \begin{gather*} \|\Id_E\tens\Fourier_\ph\|_{\borne(\Ell^p(E))} \le\|\Id_E\tens\Schur_\Hph\|_{\borne(\Sch^p[E])},\\ \|\Id_E\tens\Fourier_\ph\|_{\cb(\Ell^p(E))} =\|\Id_E\tens\Schur_\Hph\|_{\cb(\Sch^p[E])}. \end{gather*} The compression $\phi_\iota$ provides a two-sided approximation of an element $x$, whereas the proof of Theorem~\ref{trprp} uses only a one-sided approximation. This subtlety makes a difference in our second way to obtain embeddings, a direct proof by complex interpolation. \begin{prp} Let\/ $\Ga$~be a discrete amenable group and let\/ $(\mu_\iota)$~be a \emph{Reiter net of means} for\/~$\Ga$: \begin{itemize} \item each\/~$\mu_\iota$ is a positive sequence summing to\/~1 with finite support\/~$\Ga_\iota\subseteq\Ga$ and viewed as a diagonal operator from\/~$\ell^2_{\Ga_\iota}$ to\/~$\ell^2_\Ga$, so that \begin{equation*} \|\mu_\iota\|_{\Sch^1}=\sum_{\ga\in\Ga_\iota}{(\mu_\iota)}_{\ga}=1; \end{equation*} \item the net\/~$(\mu_\iota)$ satisfies, for each\/~$\ga\in\Ga$, \emph{Reiter's Property~$P_1$}: \begin{equation*} \sum_{\beta\in\Ga}{\bigl|{(\mu_\iota)}_{\ga^{-1}\beta}-{(\mu_\iota)}_{\beta}\bigr|}\to0. \end{equation*} \end{itemize} Let\/~$x\in\mat_n\otimes\vn=\vn(\tr\otimes\tau)$ and\/~$p\ge1$. Then \begin{equation*} \limsup\|x\mu_\iota^{1/p}\|_{\Sch^p_{\vphantom{n}}(\Sch^p_n)}=\|x\|_{\Ell^p(\tr\otimes\tau)}. \end{equation*} \end{prp} \begin{proof} Consider~$x=\sum_{\ga\in\Ga}x_\ga\tens\la_\ga$ with only a finite number of the~$x_\ga\in\mat_n$ nonzero. As \begin{equation*} \sum_{\beta\in\Ga}{\bigl|{(\mu_\iota)}_{\ga^{-1}\beta}^{1/2}-{(\mu_\iota)}_{\beta}^{1/2}\bigr|}^2\le \sum_{\beta\in\Ga}{\bigl|{(\mu_\iota)}_{\ga^{-1}\beta}-{(\mu_\iota)}_{\beta}\bigr|}, \end{equation*} Property~$P_1$ implies \emph{Property~$P_2$}: \begin{equation*} \|\la_\ga\mu_\iota^{1/2}-\mu_\iota^{1/2}\la_\ga\|_{\Sch^2}\to0, \end{equation*} so that \begin{equation*} \|x\mu_\iota^{1/2}-\mu_\iota^{1/2}x\|_{\Sch^2(\Sch^2_n)}\to0. \end{equation*} As the $\mat_n$-valued matrix of~$x$ for the canonical basis of~$\ell^2_\Ga$ is~$(x_{rc^{-1}})_{(r,c)\in\Ga\times\Ga}$, \begin{align*} \|x\mu_\iota^{1/2}\|_{\Sch^2(\Sch^2_n)}^2 &=\sum_{(r,c)\in\Ga\times\Ga}\|x_{rc^{-1}}\|_{\Sch^n_2}^2{(\mu_\iota)}_c\\ &=\sum_{c\in\Ga}{(\mu_\iota)}_c\sum_{r\in\Ga}\|x_{rc^{-1}}\|_{\Sch^n_2}^2\\ &=\sum_{c\in\Ga}{(\mu_\iota)}_c\|x\|_{\Ell^2(\tr\otimes\tau)}^2=\|x\|_{\Ell^2(\tr\otimes\tau)}^2. \end{align*} By density and continuity, the result extends to all~$x\in \Ell^2(\tr\otimes\tau)$. Let us prove now that for~$x\in\Ell^\infty(\tr\otimes\tau)$, \begin{equation*} \limsup\|x\mu_\iota\|_{\Sch^1(\Sch^1_n)}\le\|x\|_{\Ell^1(\tr\otimes\tau)}. \end{equation*} The polar decomposition~$x=u|x|$ yields a factorisation~$x=ab$ with~$a=u\abs{x}^{1/2}$ and~$b=\abs{x}^{1/2}$ in~$\Ell^\infty(\tr\otimes\tau)$ such that \begin{gather*} \|a\|_{\Ell^2(\tr\otimes\tau)}=\|b\|_{\Ell^2(\tr\otimes\tau)}=\|x\|_{\Ell^1(\tr\otimes\tau)}^{1/2} \\ \|a\|_{\Ell^\infty(\tr\otimes\tau)}=\|x\|_{\Ell^\infty(\tr\otimes\tau)}^{1/2}. \end{gather*} Then \begin{math} x\mu_\iota=a(b\mu_\iota^{1/2}-\mu_\iota^{1/2}b)\mu_\iota^{1/2}+a\mu_\iota^{1/2}b\mu_\iota^{1/2} \end{math}, so that the Cauchy-Schwarz inequality yields \begin{align*} \|x\mu_\iota\|_{\Sch^1(\Sch^1_n)} &\le\|a\|_{\Ell^\infty(\tr\otimes\tau)} \|(b\mu_\iota^{1/2}-\mu_\iota^{1/2}b)\mu_\iota^{1/2}\|_{\Sch^1(\Sch^1_n)} +\|a\mu_\iota^{1/2}b\mu_\iota^{1/2}\|_{\Sch^1(\Sch^1_n)}\\ &\le\|a\|_{\Ell^\infty(\tr\otimes\tau)} \|b\mu_\iota^{1/2}-\mu_\iota^{1/2}b\|_{\Sch^2(\Sch^2_n)} +\|a\|_{\Ell^2(\tr\otimes\tau)}\|b\|_{\Ell^2(\tr\otimes\tau)} \end{align*} and therefore our claim. Now complex interpolation yields \begin{equation*} \limsup\|x\mu_\iota^{1/p}\|_{\Sch^p_{\vphantom{n}}(\Sch^p_n)}\le\|x\|_{\Ell^p(\tr\otimes\tau)} \end{equation*} for~$x\in\Ell^\infty(\tr\otimes\tau)$ and~$p\in[1,\infty]$. In fact, consider the function~$f(z)=u\abs{x}^{pz}\mu_\iota^z$ analytic in the strip~$0<\Im z<1$ and continuous on its closure; then $f(\iu t)$~is a product of unitaries for~$t\in\R$, so that \begin{math} \|f(\iu t)\|_{\vn(\tr\otimes\tau)}=1. \end{math} Also \begin{align*} \|f(1+\iu t)\|_{\Sch^1(\Sch^1_n)} &=\|\abs{x}^p\mu_\iota\|_{\Sch^1(\Sch^1_n)}. \end{align*} As $\Sch^p(\Sch^p_n)$~is the complex interpolation space~$(\Sch^\infty(\mat_n),\Sch^1(\Sch^1_n))_{1/p}$, \begin{align*} \|x\mu_\iota^{1/p}\|_{\Sch^p_{\vphantom{n}}(\Sch^p_n)}= \|f(1/p)\|_{\Sch^p_{\vphantom{n}}(\Sch^p_n)} \le \||x|^p\mu_\iota\|_{\Sch^1(\Sch^1_n)}^{1/p}. \end{align*} Then, taking the upper limit and using the estimate on~$\Sch^1(\Sch^1_n)$, \begin{align*} \limsup \|x\mu_\iota^{1/p}\|_{\Sch^p_{\vphantom{n}}(\Sch^p_n)}&\le \limsup \||x|^p\mu_\iota\|_{\Sch^1(\Sch^1_n)}^{1/p}\\ &\le \||x|^p\|_{\Ell^1(\tr\otimes\tau)}^{1/p}=\|x\|_{\Ell^p(\tr\otimes\tau)}. \end{align*} The reverse inequality is obtained by duality; first note that for~$y\in{\vn(\tr\otimes\tau)}$, \begin{equation*}\lim \tr\otimes\tr(y\mu_\iota)= \tr\otimes\tau(y).\end{equation*} With the above notation and the inequality for~$p'$, \begin{align*} \|x\|_{\Ell^p(\tr\otimes\tau)}^p&=\tau (|x|^p)=\lim \tr |x|^p\mu_\iota = \lim\tr \mu_\iota^{1-1/p}|x|^{p-1}u^*x\mu_\iota^{1/p}\\ &\le \limsup \|\mu_\iota^{1-1/p}|x|^{p-1}\|_{\Sch^{p'}_{\vphantom{n}}(\Sch^{p'}_n)} \|x\mu_\iota^{1/p}\|_{\Sch^p_{\vphantom{n}}(\Sch^p_n)}\\&= \limsup \||x|^{p-1}\mu_\iota^{1-1/p}\|_{\Sch^{p'}_{\vphantom{n}}(\Sch^{p'}_n)} \|x\mu_\iota^{1/p}\|_{\Sch^p_{\vphantom{n}}(\Sch^p_n)} \\&\le \||x|^{p-1}\|_{\Ell^{p'}(\tr\otimes\tau)}\limsup\|x\mu_\iota^{1/p}\|_{\Sch^p_{\vphantom{n}}(\Sch^p_n)}, \end{align*} so that \begin{equation*} \limsup\|x\mu_\iota^{1/p}\|_{\Sch^p_{\vphantom{n}}(\Sch^p_n)}=\|x\|_{\Ell^p(\tr\otimes\tau)}^p.\qedhere \end{equation*} \end{proof} \begin{rem}Let $\mu$~be any positive diagonal operator with $\tr \mu=1$ and $p\ge 2$; then $\|x\mu^{1/p}\|_{\Sch^p(\Sch^p_n)}\le \|x\|_{\Ell^p}$ for all $x\in \vn(\tr\otimes\tau)$. The Reiter condition is only necessary to go below exponent~2. \end{rem} We could also have used interpolation with a two-sided approximation by Reiter means. We would have obtained \begin{equation*} \limsup\|\mu_\iota^{1/2p}x\mu_\iota^{1/2p}\|_{\Sch^p_{\vphantom{n}}(\Sch^p_n)} =\|x\|_{\Ell^p(\tr\otimes\tau)}. \end{equation*} This formula is in the spirit of the first approach of this section. \section{Transfer of lacunary sets into lacunary matrix patterns} \label{sec:lac} As a first application of Theorem~\ref{trans2}, let us mention that it provides a shortcut for some arguments in \cite{ha98}, as it permits us to transfer lacunary subsets of a discrete group $\Ga$ into lacunary matrix patterns in $\Ga\times\Ga$. Let us first introduce the following terminology. \begin{dfn}\label{unclp:def} Let $\Ga$~be a discrete group and~$\La\subseteq\Ga$. Let $X$~be the reduced $\Cont^*$-algebra $\Cont$ of~$\Ga$ or its noncommutative Lebesgue space~$\Ell^p$ for~$p\in\iod[1,\infty]$. \begin{enumerate} \item The set~$\La$ is \emph{unconditional} in~$X$ if the Fourier series of every~$x\in X_\La$ converges unconditionally; i.e., there is a constant~$D$ such that \begin{equation*} \biggnorm{\sum_{\ga\in\La'}x_\ga\eps_\ga\la_\ga}_{X} \le D\|x\|_{X} \end{equation*} for finite~$\La'\subseteq\La$ and~$\eps_\ga\in\T$. The minimal constant~$D$ is the \emph{unconditional constant of~$\La$ in~$X$}. \item If~$X=\Cont$, let~$\tilde X=\Sch^\infty\otimes\Cont$; if~$X=\Ell^p$, let~$\tilde X=\Ell^p(\tr\otimes\tau)$. The set~$\La$ is \emph{completely unconditional} in~$X$ if the Fourier series of every~$x\in\tilde X_\La$ converges unconditionally; i.e., there is a constant~$D$ such that \begin{equation*} \biggnorm{\sum_{\ga\in\La'}x_\ga\otimes\eps_\ga\la_\ga}_{\tilde X} \le D\|x\|_{\tilde X} \end{equation*} for finite~$\La'\subseteq\La$ and~$\eps_\ga\in\T$. The minimal constant~$D$ is the \emph{complete unconditional constant of~$\La$ in~$X$}. \end{enumerate} \end{dfn} Unconditional sets in~$\Ell^p$ have been introduced as $\Lambda(p)$ sets'' in \cite[Definition~1.1]{ha98} for $p>2$. If $\Ga$~is abelian, they are Walter Rudin's $\Lambda(p)$ sets if $p>2$ and his $\Lambda(2)$ sets if $p<2$ (see \cite{ru60,bo01}). Asma Harcharras (\cite[Definition~1.5, Comments~1.9]{ha98}) called completely unconditional sets in~$\Ell^p$ $\Lambda(p)_{\cb}$ sets'' if~$p\in\io[2,\infty]$, and $\mathrm{K}(p)_{\cb}$ sets'' if~$p\in\iog[1,2]$; her definitions are equivalent to ours by the noncommutative Khinchin inequality. Sets that are unconditional in $\Cont$ have been introduced as unconditional Sidon sets'' in \cite{bo81a}. If $\Ga$ is amenable, Fourier multipliers are automatically c.b.\ on $\Cont_\La$, so that such sets are automatically completely unconditional in~$\Cont$, and there are at least three more equivalent definitions for the counterpart of Sidon sets in an abelian group. If $\Ga$ is nonamenable, these definitions are no longer all equivalent, and our notion of completely unconditional sets in~$\Cont$ corresponds to Marek Bo\.zejko's c.b.\ Sidon sets.'' \begin{dfn} \label{uncsp:def} Let $1\le p\le\infty$ and $I$ be a subset of the product $\Row\times\Col$ of two index sets. \begin{enumerate} \item The set~$I$ is \emph{unconditional} in the Schatten-von-Neumann class~$\Sch^p$ associated with $\borne(\ell^2_\Col,\ell^2_R)$ if the matrix representation of every~$x\in\Sch^p_I$ converges unconditionally; i.e., there is a constant~$D$ such that \begin{equation*} \Bigl\|\sum_{q\in I'}x_q\eps_q\e_q\Bigr\|_p \le D \|x\|_p \end{equation*} for finite~$I'\subseteq I$ and~$\eps_q\in\T$. The minimal constant~$D$ is the \emph{unconditional constant of $I$ in $\Sch^p$}. \item The set $I$ is \emph{completely} unconditional in $\Sch^p$ if the matrix representation of every~$x\in\Sch^p_I(\Sch^p_{\vphantom{I}})$ converges unconditionally; i.e., there is a constant~$D$ such that \begin{equation*}\label{cu} \Bigl\|\sum_{q\in I'}x_q\tens\eps_q\e_q\Bigr\|_p \le D \|x\|_p \end{equation*} for finite~$I'\subseteq I$ and~$\eps_q\in\T$. The minimal constant~$D$ is the \emph{complete unconditional constant of $I$ in $\Sch^p$}. \end{enumerate} \end{dfn} Harcharras called unconditional and completely unconditional sets in~$\Sch^p$ $\sigma(p)$ sets'' and $\sigma(p)_{\cb}$ sets'', respectively (\cite[Definitions 4.1~and~4.4, Remarks~4.6$\,(iv)$]{ha98}); she supposed $p<\infty$, so that her definitions are equivalent to ours by the noncommutative Khin\-chin inequality. \begin{prp}\label{trans:set} Let\/ $\Ga$~be a discrete group. Let\/ $\La\subseteq\Ga$ and consider the associated Toeplitz set\/ $\HLa=\{(r,c)\in\Ga\times\Ga:\row\col^{-1}\in \La\}$. Let\/ $p\in\iod[1,\infty]$. \begin{enumerate} \item If\/ $\Ga$ is amenable, then\/ $\La$ is unconditional in\/ $\Ell^p$ if\/ $\HLa$ is unconditional in\/ $\Sch^p$. \item If\/ $\La$ is completely unconditional in\/ $\Ell^p$, then\/ $\HLa$ is completely unconditional in\/ $\Sch^p$. The converse holds if\/ $\Ga$ is amenable. \end{enumerate} \end{prp} \begin{proof} The first part of $(b)$ follows by the argument of the proof of \cite[Proposition~4.7]{ha98}; let us sketch it. Consider the isometric embedding of the space $\Sch^p_{\mkern-3mu\HLa}(\Sch^p)$ in $\Ell^p_\La(\tr\otimes\tr\otimes\tau)$ that is given in the proof of Lemma~\ref{trans1} and apply the equivalent Definition~1.5 in \cite{ha98} of the complete unconditionality of~$\La$: this gives the complete unconditionality of~$\HLa$ in the equivalent Definition~4.4 in \cite{ha98}. Unconditionality in $\Ell^p$ expresses the uniform boundedness of relative unimodular Fourier multipliers on $\Ell^p_\La$; complete unconditionality expresses their uniform complete boundedness. Unconditionality in $\Sch^p$ expresses the uniform boundedness of relative unimodular Schur multipliers on $\Sch^p_{\mkern-3mu\HLa}$; complete unconditionality expresses their uniform complete boundedness. The second part of $(b)$ follows therefore from Theorem~\ref{trans2}$\,(b)$ and $(a)$ follows from Theorem~\ref{trprp}. \end{proof} \begin{rem} This transfer does not pass to the limit $p=\infty$ in~$(b)$ and is void in~$(a)$. Nicholas Varopoulos proved that unconditional sets in $\Sch^\infty$ are finite unions of patterns whose rows or whose columns contain at most one element, and this excludes sets of the form $\HLa$ for any infinite $\La$ (\cite[Theorem~4.2]{va69}, see \cite[\S\,5]{ne06} for a reader's guide). \end{rem} \begin{rem} See \cite[Remark~11.3]{ne06} for an illustration of Proposition~4.3$\,(b)$ in a particular context. \end{rem} \begin{rem} Let $p$ be an even integer greater than or equal to 4. The existence of a $\sigma(p)_\cb$ set that is not a $\sigma(q)$ set for any $q>p$ (\cite[Theorem~4.9]{ha98}) becomes a direct consequence of Walter Rudin's construction (\cite[Theorem~4.8]{ru60}) of a $\Lambda(p)$ set that is not a $\Lambda(q)$ set for any $q>p$, because this set has property $\mathrm B(p/2)$ (\cite[Definition~2.4]{ha98}) and is therefore $\Lambda(p)_\cb$ by \cite[Theorem~1.13]{ha98} (in fact, it is even 1-unconditional'' in $\Ell^p$ because $\mathrm B(p/2)$ is $p/2$-independence'' (\cite[\S\,11]{ne06})). \end{rem} \begin{rem} In the same way, \cite[Theorem~5.2]{ha98} becomes a mere reformulation of \cite[Proposition 3.6]{ha98} if one remembers that the Toeplitz Schur multipliers are 1-complemented in the Schur multipliers for an amenable discrete group and for all classical norms. Basically, results on $\Lambda(p)_{\cb}$ sets produce results on $\sigma(p)_{\cb}$ sets. \end{rem} Let us now estimate the complete unconditional constant of sumsets. In the case~$\Ga=\Z$, Harcharras (\cite[Prop.~2.8]{ha98}) proved that a completely unconditional set in~$\Ell^p$ cannot contain the sumset of characters~$A+A$ for arbitrary large finite sets~$A$. In particular, if $\La\supseteq A+A$ with $A$ infinite, then $\La$~is not a completely unconditional set in $\Ell^p$. Thus, her proof provided examples of~$\Lambda(p)$ sets that are not $\Lambda(p)_\cb$ sets. We generalise Harcharras' result in two directions. Compare~\cite[\S\,1.4]{lr75}. \begin{prp}\label{har:lp} Let\/ $\Ga$~be a discrete group and\/~$p\ne2$. A completely unconditional set in\/~$\Ell^p$ cannot contain the sumset of two arbitrarily large sets. More precisely, let\/ $\Row$~and\/~$\Col$ be subsets of\/~$\Ga$ with\/~$\card{\Row}\ge n$ and\/ $\card{\Col}\ge n^3$. Then, for any\/~$p\ge1$, the complete unconditional constant of the sumset\/~$\Row\Col$ in\/~$\Ell^p$ is at least\/~$n^{|1/2-1/p|}$. \end{prp} \begin{proof} Let $r_1,\dots,r_{n}$ be pairwise distinct elements in $\Row$. We shall select inductively elements $c_1,\dots,c_{n}$ in $\Col$ such that the $r_ic_j$ are pairwise distinct. Assume there are $c_1,\dots,c_{m-1}$ such that the induction hypothesis \begin{equation*} \forall\, i,k\le n\ \forall\,j,l\le m-1\quad (i,j)\ne(k,l)\ \imp\ \row_i\col_j\ne\row_k\col_l. \end{equation*} holds. We are looking for an element $c_m\in\Col$ such that \begin{equation*} \forall\, i,k\le n\ \forall\, l\le m-1\quad \row_i\col_m\ne\row_k\col_l. \end{equation*} Such an element exists as long as $m\le n$, because the set $\{r_i^{-1}r_kc_l:i,k\le n,\,\allowbreak l\le m-1\}$ has at most $\bigl(n(n-1)+1\bigr)(m-1)j\}$~does not form a complete~$\Lambda(p)$ set for any~$p\ne2$. Indeed, \ $\{2^i-2^j\}=\La\cup-\La$ does not, and if $\La$ did, then so would $-\La$ and~$\La\cup-\La$. \end{exa} \section{\texorpdfstring{Toeplitz Schur multipliers on~$\Sch^p$ for~$p<1$}{Toeplitz Schur multipliers on Sp for p<1}}\label{sec:tsp1} When~$0j$. Then the norm of the Schur multiplier\/~$\rh$ on\/~$\Sch^\psi$ coincides with the norm of the Schur multiplier\/~$\rh$ on\/ $\Sch^\psi(\Sch^\psi)$. \end{prp} \begin{proof} Let $a\in\Sch^\psi_m(\Sch^\psi_n)$; \ $a$~may be considered as an~$m\times m$ matrix~$(a_{ij})$ whose entries~$a_{ij}$ are $n\times n$ matrices, and may be identified with the block matrix \begin{equation*} \tilde{a}= \begin{pmatrix} 0&a_{11}&0&a_{12}&\cdots\\ 0&0&0&0&\cdots\\ 0&a_{21}&0&a_{22}&\cdots\\ 0&0&0&0&\cdots\\ \vdots&\vdots&\vdots&\vdots&\ddots \end{pmatrix}. \end{equation*} In this identification, \ $\Id_{\Sch^\psi_n}\otimes\Schur_\rh(a)$ is~$\Schur_\rh(\tilde{a})$. \end{proof} The Hilbert transform~$\Ht$ is the Schur multiplier obtained by choosing $z=-1$ and $w=1$. The upper triangular operators in~$\Sch^p$ can be seen as a noncommutative $\Hardy^p$ space, and $\Ht$ corresponds exactly to the Hilbert transform in this setting (see \cite{Ran98,MaWe98}). Using classical results on $\Hardy^p$ spaces, all Hilbert transforms are c.b.\ for~$12$ by this method. The Riesz projection~$\Rt$ is the Schur multiplier obtained by choosing $z=0$ and $w=1$ in Proposition~\ref{rhcb}. It is the projection on the upper triangular part. On the circle, the classical Riesz projection~$T$, that is the projection onto the analytic part, corresponds to the Fourier multiplier given by the indicator function~$\un_{\Z^+}$ of nonnegative integers; its norm on~$\Ell^p$, as computed by Hollenbeck and Verbitsky (\cite{HoVe00}), is \begin{math} \csc(\pi/p)\end{math}. As for the Hilbert transform, we know that the norm and the complete norm of~$\Rt$ on~$\Sch^p$ are equal and coincide with the complete norm of~$T$ on~$\Ell^p$, but, to the best of our knowledge, there is no simple formula like~\eqref{cotlar} to go from exponent~$p$ to~$2p$. We only obtained the following computation. \begin{prp}\label{normt} Let\/ $p\in\io[1,\infty]$. The norm and the complete norm of the Riesz projection\/~$\Rt$ on\/~$\Sch^p$ coincide with the complete norm of the Riesz projection\/~$T$ on\/~$\Ell^p$: if\/ $\Hun(i,j)=\un_{\Z^+}(i-j)$ for\/~$i,j\ge1$, \begin{equation*} \|\Hun\|_{\Schur(\Sch^p)}=\|\Hun\|_{\Schur_\cb(\Sch^p)}=\|\un_{\Z^+}\|_{\Fourier_\cb(\Ell^p)}. \end{equation*} If\/~$p=4$, then these norms coincide with the norm of\/~$T$ on\/~$\Ell^p$: \begin{equation*} \|\Hun\|_{\Schur(\Sch^4)}=\|\Hun\|_{\Schur_\cb(\Sch^4)}=\|\un_{\Z^+}\|_{\Fourier_\cb(\Ell^4)}=\|\un_{\Z^+}\|_{\Fourier(\Ell^4)}=\sqrt2. \end{equation*} \end{prp} \begin{proof} We shall compute the norm of~$\Rt$ on~$\Sch^4$. Let $x$~be a finite upper triangular matrix and let $y$~be a finite strictly lower triangular matrix. We have to prove that \begin{equation*} \sqrt 2 \|x+y\|_{\Sch^4}\ge \|x\|_{\Sch^4}. \end{equation*} Let us make the obvious estimates on~$\Sch^2$ and use the fact that the adjoint operation is isometric: \begin{equation*} \|\Rt (xx^*)\|_{\Sch^2} =\|\Rt ((x+y)x^*)\|_{\Sch^2} \le\|x+y\|_{\Sch^4}\|x\|_{\Sch^4}, \end{equation*} and similarly, \begin{equation*} \| (\Id-\Rt) (xx^*)\|_{\Sch^2}=\| (\Id -\Rt) (x(x+y)^*)\|_{\Sch^2} \le \|x\|_{\Sch^4} \|x+y\|_{\Sch^4}. \end{equation*} As $\Rt$~and~$\Id-\Rt$ have orthogonal ranges, \begin{equation*} \|x\|_{\Sch^4}^4=\|xx^*\|_{\Sch^2}^2=\| (\Id-\Rt) (xx^*)\|_{\Sch^2}^2+ \| \Rt (xx^*)\|_{\Sch^2}^2\le 2 \|x\|_{\Sch^4}^2 \|x+y\|_{\Sch^4}^2.\qedhere \end{equation*} \end{proof} \section{Unconditional approximating sequences}\label{sec:uap} The following definition makes sense for general operator spaces, but we choose to state it only in our specific context. \begin{dfn}\label{uap:def} Let $\Ga$~be a discrete group and~$\La\subseteq\Ga$. Let $X$~be the reduced $\Cont^*$-algebra of~$\Ga$ or its noncommutative Lebesgue space~$\Ell^p$ for~$p\in\iod[1,\infty]$. \begin{enumerate} \item A sequence~$(T_k)$ of operators on~$X_\La$ is an \emph{approximating sequence} if each~$T_k$ has finite rank and~$T_kx\to x$ for every~$x\in X_\La$. It is a \emph{complete} approximating sequence if the~$T_k$ are uniformly c.b. If $X_\La$ admits a complete approximating sequence, then $X_\La$ enjoys the \emph{c.b.\ approximation property}. \item The \emph{difference sequence}~$(\Delta T_k)$ of a sequence~$(T_k)$ is given by~$\Delta T_1=T_1$ and $\Delta T_k=T_k-T_{k-1}$ for~$k\ge2$. An approximating sequence~$(T_k)$ is \emph{unconditional} if the operators \begin{equation} \label{cuap:eq} \sum_{k=1}^n\eps_k\Delta T_k\quad\text{with~$n\ge1$ and~$\eps_k\in\{-1,1\}$} \end{equation} are uniformly bounded on~$X_\La$; then~$X_\La$ enjoys the \emph{unconditional approximation property}. \item An approximating sequence~$(T_k)$ is \emph{completely} unconditional if the operators in~\eqref{cuap:eq} are uniformly c.b.\ on~$X_\La$; then $X_\La$ enjoys the \emph{complete} unconditional approximation property. The minimal uniform bound of these operators is the \emph{complete unconditional constant} of~$X_\La$. \end{enumerate} \end{dfn} We may always suppose that a complete approximating sequence on~$\Cont_\La$ is a Fourier multiplier sequence (see \cite[Theorem~2.1]{hk94}). We may also do so on~$\Ell^p_\La$ if $\Ell^\infty$ has the so-called QWEP (see \cite[Theorem~4.4]{jr03}). More precisely, the following proposition holds. \begin{prp} \label{fm} Let\/ $\Ga$ be a discrete group and\/ $\La\subseteq\Ga$. Let\/ $X$~either be its reduced\/ $\Cont^*$-algebra or its noncommutative Lebesgue space\/~$\Ell^p$, where\/ $p\in\iod[1,\infty]$ and\/ $\Ell^\infty$ has the QWEP\@. If\/ $X_\La$ enjoys the completely unconditional approximation property with constant\/~$D$, then for every\/~$D'>D$ there is a complete approximating sequence of Fourier multipliers\/~$(\ph_k)$ that realises the completely unconditional approximation property with constant\/~$D'$: the Fourier multipliers\/~$\sum_{k=1}^n\eps_k\Delta\ph_k$ are uniformly completely bounded by\/~$D'$ on\/~$X_\La$. \end{prp} Let us now describe how to skip blocks in an approximating sequence in order to construct an operator that acts like the Riesz projection on the sumset of two infinite sets. The following trick will be used in the induction below (compare with the proof of \cite[Theorem~4.2]{nptv}): \begin{equation*} \begin{pmatrix} 1&1\hskip\arraycolsep\vrule\hskip\arraycolsep0\\0&1\hskip\arraycolsep\vrule\hskip\arraycolsep0\\\noalign{\hrule}0&0\hskip\arraycolsep\vrule\hskip\arraycolsep0 \end{pmatrix} - \begin{pmatrix} 1&1\hskip\arraycolsep\vrule\hskip\arraycolsep0\\1&1\hskip\arraycolsep\vrule\hskip\arraycolsep0\\1&1\hskip\arraycolsep\vrule\hskip\arraycolsep0 \end{pmatrix} + \begin{pmatrix} 1&1&1\\1&1&1\\1&1&1 \end{pmatrix} = \begin{pmatrix} 1&1&1\\0&1&1\\0&0&1 \end{pmatrix}. \end{equation*} \begin{lem} \label{diag} Let\/ $\Ga$~be a discrete group and\/~$\La\subseteq\Ga$. Suppose that\/ $\La$ contains the sumset\/~$\Row\Col$ of two infinite sets\/ $\Row$~and\/~$\Col$. Let\/ $(T_k)$~be either an approximating sequence on\/~$\Ell^p_\La$ with\/ $p\in\iod[1,\infty]$, or an approximating sequence of Fourier multipliers on\/~$\Cont_\La$. Let\/~$\eps>0$. There is a sequence\/~$(\row_i)$ in\/~$\Row$, a sequence\/~$(\col_i)$ in\/~$\Col$, and there are indices\/~$l_10$ be chosen later. \begin{itemize} \item The operator~$U_n$ defined by Equation~\eqref{sbs} has finite rank. If it is a Fourier multiplier, one can choose an element~$\row_{n+1}\in \Row$ such that $U_n(\la_{\row_{n+1}\col_j})=0$ for~$j\le n$. If it acts on~$\Ell^p_\La$ with $p\in\iod[1,\infty]$, one can choose an element~$\row_{n+1}\in \Row$ such that $\norm{U_n(\la_{\row_{n+1}\col_j})}<\delta$ for~$j\le n$ because $(\la_\ga)_{\ga\in\Ga}$ is weakly null in $\Ell^p$. \item There is~$k_{n+1}>l_n$ such that $\norm{T_{k_{n+1}}(\la_\ga)-\la_\ga}<\delta$ for~$\ga\in\{\row_i\col_j:1\le i\le n+1,1\le j\le n\}$. \item Again, choose~$\col_{n+1}\in \Col$ such that $\norm{(U_n-T_{k_{n+1}})(\la_{\row_i\col_{n+1}})}<\delta$ for~$i\le n+1$. \item Again, choose~$l_{n+1}>k_{n+1}$ such that $\norm{T_{l_{n+1}}(\la_\ga)-\la_\ga}<\delta$ for~$\ga\in\{\row_i\col_j:1\le i,j\le n+1\}$. \end{itemize} Let~$U_{n+1}=U_n+(T_{l_{n+1}}-T_{k_{n+1}})$. If $i\le n+1$ and~$j\le n$, then \begin{equation*} \norm{\Delta U_{n+1}(\la_{r_ic_j})}\le\norm{T_{l_{n+1}}(\la_{r_ic_j})-\la_{r_ic_j}}+\norm{\la_{r_ic_j}-T_{k_{n+1}}(\la_{r_ic_j})}<2\delta, \end{equation*} so that \begin{align*} \norm{U_{n+1}(\la_{r_ic_j})-\la_{r_ic_j}}<\eps+2\delta&\quad\text{if $i\le j\le n$}\\ \norm{U_{n+1}(\la_{r_ic_j})}<\eps+2\delta&\quad\text{if $j0$ and every~$n$, there are elements~$r_1,\dots,r_n\in\Row$, $c_1,\dots,c_n\in\Col$ such that the Fourier multiplier~$\ph$ given by the indicator function of ${\{r_ic_j\}_{i\le j}}$ is near to a skipped block sum~$U_n$ of~$(T_k)$ in the sense that $\|U_n(\la_{r_ic_j})-\ph_{r_ic_j}\la_{r_ic_j}\|<\eps$. But $U_n$~is the mean of two operators of the form~\eqref{cuap:eq}: its complete norm will provide a lower bound for the complete unconditional constant of~$X_\La$. Let us repeat the argument of Lemma~\ref{trans1} with~$x\in\Sch^p_n$. As \begin{multline*} \Bignorm{\sum_{i,j=1}^nx_{i,j}\e_{i,j}}_{\Sch^p_n}\\ \begin{aligned} &=\Bignorm{\Bigl(\sum_{i=1}^n\e_{i,i}\otimes\la_{r_i}\Bigr)\Bigl(\sum_{i,j=1}^nx_{i,j}\e_{i,j}\otimes\la_\id\Bigr)\Bigl(\sum_{j=1}^n\e_{j,j}\otimes\la_{c_j}\Bigr)}_{\Ell^p(\tr\otimes\tau)}\\ &=\Bignorm{\sum_{i=1}^nx_{i,j}\e_{i,j}\otimes\la_{r_ic_j}}_{\Ell^p(\tr\otimes\tau)} \end{aligned} \end{multline*} and \begin{equation*} \Bignorm{\sum_{i=1}^nx_{i,j}\e_{i,j}\otimes(U_n(\la_{r_ic_j})-\ph_{r_ic_j}\la_{r_ic_j})}_{\Ell^p(\tr\otimes\tau)}\Bignorm{\sum_{i\le j}x_{i,j}\e_{i,j}\otimes\la_{r_ic_j}}_{\Ell^p(\tr\otimes\tau)}-n^2\eps\norm{x}_{\Sch^p_n}\\ &=\norm{\Rt(x)}_{\Sch^p_n}-n^2\eps\norm{x}_{\Sch^p_n}. \end{align*} This proves~$(a)$ as well as the first assertion in~$(b)$, because the Riesz projection is unbounded on~$\Sch^1$. Let $(T_k)$~be an approximating sequence on~$\Cont_\La$; by Lemma~\ref{fm}, we may suppose that $(T_k)$~is a sequence of Fourier multipliers. Thus the second assertion in~$(b)$ follows from Lemma~\ref{diag} combined with the preceding argument (where $\Sch^p_n$~is replaced by~$\mat_n$ and $\Ell^p(\tr\otimes\tau)$ by~$\mat_n\otimes\Cont$) and the unboundedness of the Riesz projection on~$\Sch^\infty$. For~$(c)$, note that the Fourier multipliers $T_k$ are automatically c.b.\ on~$\Cont_\La$ if $\Ga$~is amenable (proof of Theorem~\ref{trans2}$\,(c)$). \end{proof} Theorem~\ref{T}$\,(b)$ was originally devised to prove that the Hardy space $\mathrm H^1$, corresponding to the case $\La=\N\subseteq\Z$ and $p=1$, admits no completely unconditional basis (see \cite{ri00,ri01}). Theorem~\ref{T}$\,(c)$ both generalises the fact that a sumset cannot be a Sidon set (see \cite[\S\S\,1.4,\,6.6]{lr75} for two proofs and historical remarks, or \cite[Proposition~IV.7]{lq04}) and Daniel Li's result \cite[Corollary~13]{li96} that the space~$\Cont_\La$ does not have the metric'' unconditional approximation property if $\Ga$~is abelian and $\La$ contains a sumset. Li (\cite[Theorem~10]{li96}) also constructed a set $\La\subseteq\Z$ such that $\Cont_\La$ has this property, while $\La$ contains the sumset of arbitrarily large sets. This theorem also provides a new proof that the disc algebra has no unconditional basis and answers \cite[Question~6.1.6]{ne99}. \begin{exa} Neither the span of products~$\{r_ir_j\}$ of two Rademacher functions in the space of continuous functions on~$\{-1,1\}^\infty$ nor the span of products~$\{s_is_j\}$ of two Steinhaus functions in the space of continuous functions on~${\T}^\infty$ has an unconditional basis. \end{exa} \section{Relative Schur multipliers of rank one}\label{sec:rsm1} Let $\rh$ be an \emph{elementary} Schur multiplier on $\Sch^\infty$, that is, $\rh=x\otimes y=(x_ry_c)_{(\row,\col)\in\Row\times\Col}.$ Then its norm is $\sup_{\row\in\Row}\abs{x_\row}\sup_{\col\in\Col}\abs{y_\col}$. How is this norm affected if $\rh$ is only partially specified, that is, if the action of~$\rh$ is restricted to matrices with a given support? \begin{thm} \label{mps} Let\/ $I\subseteq\Row\times\Col$ and consider\/ $(x_\row)_{\row\in\Row}$~and\/~$(y_\col)_{\col\in\Col}$. The relative Schur multiplier on\/~$\Sch^\infty_I$ given by\/ $(x_\row y_\col)_{(\row,\col)\in I}$ has norm\/~$\sup_{(\row,\col)\in I}\abs{x_\row y_\col}$. \end{thm} Note that the norm of the Schur multiplier $(x_\row y_\col)_{(\row,\col)\in I}$ is bounded by \begin{math} \sup_{\row\in\Row}\abs{x_\row}\*\sup_{\col\in\Col}\abs{y_\col} \end{math} because the matrix $(x_\row y_\col)_{(\row,\col)\in\Row\times\Col}$ is a trivial extension of $(x_\row y_\col)_{(\row,\col)\in I}$; the proof below provides a constructive nontrivial extension of this Schur multiplier that is a composition of ampliations of the Schur multiplier in the following lemma. \begin{lem} \label{schur} The Schur multiplier\/ \begin{math} \begin{pmatrix} \overline z&w\\ \overline w&z\end{pmatrix} \end{math} has norm\/~$\max(\abs{z},\abs{w})$ on\/~$\mat_2$. \end{lem} \begin{proof} This follows from the decomposition \begin{equation*} \begin{pmatrix} \overline z&w\\ \overline w&z\end{pmatrix} =\frac{\abs{z}+\abs{w}}2 \begin{pmatrix} \bar tu\\t\bar u \end{pmatrix} \otimes \begin{pmatrix} \overline{tu}&tu \end{pmatrix} +\frac{\abs{z}-\abs{w}}2 \begin{pmatrix} \bar tu\\-t\bar u \end{pmatrix} \otimes \begin{pmatrix} \overline{tu}&-tu \end{pmatrix}, \end{equation*} where $t,u\in\T$ are chosen so that $z=\abs{z}t^2$ and $w=\abs{w}u^2$. \end{proof} \begin{proof}[Proof of Theorem~\ref{mps}] We may suppose that $\Col$~is the finite set~$\{1,\dots,m\}$ and that $\Row$~is the finite set~$\{1,\dots,n\}$, that each~$y_\col$ is nonzero, and that each row in~$\Row$ contains an element of~$I$. We may also suppose that $(\abs{x_\row})_{\row\in\Row}$~and~$(\abs{y_\col})_{\col\in\Col}$ are nonincreasing sequences. For each~$\row\in\Row$ let $c_r$~be the least column index of elements of~$I$ in or above row~$\row$; in other words, \begin{equation*} \col_\row=\min_{\row'\le\row}\min\{\col:(\row',\col)\in I\}. \end{equation*} The sequence~$(\col_\row)_{\row\in\Row}$ is nonincreasing. Let us define its inverse $(\row_\col)_{\col\in\Col}$ in the sense that $\row_\col\le\row\Leftrightarrow\col_\row\le\col$. For each~$\col\in\Col$, let $\row_\col=\min\{\row:\col_\row\le\col\}$. Given~$\row$, let $\row'\le\row$~be such that $(\row',\col_\row)\in I$; then $\abs{x_\row y_{\col_\row}}\le\abs{x_{\row'}y_{\col_\row}}$, so that $\sup_{\row\in\Row}\abs{x_\row y_{\col_\row}}\le\sup_{(\row,\col)\in I}\abs{x_\row y_\col}$ and the rank~1 Schur multiplier $\rh_0=(x_\row y_{\col_\row})_{(\row,\col)\in\Row\times\Col}$ with pairwise equal columns is bounded by~$\sup_{(\row,\col)\in I}\abs{x_\row y_\col}$ on~$\Sch^\infty_n$. We will now correct''~$\rh_0$ without increasing its norm so as to make it an extension of~$(x_\row y_\col)_{(\row,\col)\in I}$. Let $\row\in\Row$~and~$\col'\ge\col_\row$; then \begin{align*} x_\row y_{\col'} =x_\row y_{\col_\row}\frac{y_{\col_\row+1}}{y_{\col_\row}}\cdots\frac{y_{\col'}}{y_{\col'-1}} &=x_\row y_{\col_\row}\prod_{\col_\row\le\col\le\col'-1}\frac{y_{\col+1}}{y_{\col}}\\ &=x_\row y_{\col_\row}\prod_{\substack{\row\ge\row_\col\\\col'\ge\col+1}}\frac{y_{\col+1}}{y_{\col}}. \end{align*} This shows that it suffices to compose the Schur multiplier~$\rh_0$ with the $m-1$ rank~2 Schur multipliers with block matrix \begin{equation*} \rh_{\col}= \bordermatrix{% &\scriptstyle1~\cdots~\col&&\scriptstyle\col+1~\cdots~m\cr \begin{matrix}\scriptstyle1\\\scriptstyle\vdots\\\scriptstyle\row_\col-1\end{matrix} &\displaystyle\overline{\left(\frac{y_{\col+1}}{y_\col}\right)}&\vrule&1\cr\noalign{\kern-1pt} &\multispan3\hrulefill\cr\noalign{\kern-1pt} \hfil\begin{matrix}\scriptstyle\row_\col\\\scriptstyle\vdots\\\scriptstyle n\end{matrix} &1&\vrule&\displaystyle\frac{y_{\col+1}}{y_\col}\cr }, \end{equation*} each of which has norm~1 on~$\Sch^\infty_n$ by Lemma~\ref{schur}. \end{proof} \begin{rem} We learned after submitting this article that Timur Oikhberg proved Theorem~\ref{mps} independently and gave some applications to it; see~\cite{oi10}. \end{rem} \begin{rem} As an illustration, let $\Col=\Row=\{1,\dots,n\}$ and~$I=\{(r,c):r\ge c\}$, and let $a_i$~be an increasing sequence of positive numbers. Take $x_r=a_r$~and~$y_c=1/a_c$. Then the relative Schur multiplier~$(a_r/a_c)_{r\le c}$ has norm~1. The above proof actually constructs the norm~1 extension $\left(\min(a_r/a_c,a_c/a_r)\right)_{(r,c)}$. If we put~$a_i=\e^{x_i}$, we recover that $(\e^{-|x_r-x_c|})_{(r,c)}$~is positive definite, that is, $\abs{\cdot}$~is a conditionally negative function on~$\R$. \end{rem}\medskip \noindent 2010 \emph{Mathematics subject classification}: Primary 47B49; Secondary 43A22, 43A46, 46B28.\medskip \noindent \emph{Key words and phrases}: Fourier multiplier, Toeplitz Schur multiplier, lacunary set, unconditional approximation property, Hilbert transform, Riesz projection. %\bibliography{references-href} \begin{thebibliography}{10} \bibitem{AlPe02} A.~B. Aleksandrov and V.~V. Peller. \newblock \href{http://dx.doi.org/10.1007/s00208-002-0339-z}{Hankel and {T}oeplitz-{S}chur multipliers}. \newblock {\em Math. 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