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\title{Cycles and 1-un\-con\-di\-tio\-nal matrices}
\author{Stefan Neuwirth}
\date{}


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% \hyphenation{ana-logon ap-pli-ca-tion ap-pro-xi-ma-ting
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\begin{document}
\maketitle
\begin{abstract}
  \noindent We characterise the 1-unconditional subsets
  $(\mathrm{e}_{rc})_{(r,c) \in I}$ of the set of elementary matrices
  in the Schatten-von-Neumann class $\mathrm{S}^p$. The set of couples
  $I$ must be the set of edges of a bipartite graph without cycles of
  even length $4 \le p$ if $p$ is an even integer, and without
  cycles at all if $p$ is a positive real number that is not an even
  integer. In the latter case, $I$ is even a Varopoulos set of
  V-interpolation of constant 1. We also study the metric
  unconditional approximation property for the space $\mathrm{S}^p_I$
  spanned by $(\mathrm{e}_{rc})_{(r,c) \in I}$ in $\mathrm{S}^p$.
\end{abstract}\medskip

\renewcommand{\abstractname}{R\'esum\'e en fran\c cais}
\begin{abstract}
\noindent  Je caract{\'e}rise les sous-suites 1-inconditionnelles $(\mathrm{e}_{rc})_{(r,c)
  \in I}$ de la suite des matrices {\'e}l{\'e}mentaires dans la classe de
Schatten-von-Neumann $\mathrm{S}^p$. L'ensemble de couples $I$ doit {\^e}tre
l'ensemble des ar{\^e}tes d'un graphe biparti sans cycle de longueur paire
$l\in\{4,6,\dots,p\}$ si $p$ est un entier pair, et sans cycle du tout si $p$
est un r{\'e}el positif qui n'est pas un entier pair. Dans ce dernier cas, $I$
est m{\^e}me un ensemble de Varopoulos de V-interpolation de constante 1.
J'{\'e}tudie aussi la propri{\'e}t{\'e} d'approximation inconditionnelle m{\'e}trique
pour le sous-espace vectoriel ferm{\'e} $\mathrm{S}^p_I$ engendr{\'e} par
$(\mathrm{e}_{rc})_{(r,c) \in I}$ dans $\mathrm{S}^p$.
\end{abstract}

\section{Introduction}

\noindent The starting point for this investigation has been the following
isometric question on the Schatten-von-Neumann class $\Sch^p$.

\begin{Q}\label{q1}
  Which matrix coefficients of an operator $x\in \Sch^p$ must vanish so
  that the norm of $x$ does not depend on the argument, or on the
  sign, of the remaining nonzero matrix coefficients?
\end{Q}

Let $\Col$ be the set of columns and $\Row$ be the set of rows for
coordinates in the matrix. Let $I\subseteq\Row\times\Col$ be the set of
matrix coordinates of the nonzero matrix coefficients of $x$ (the \emph{pattern}.)
Question~\ref{q1} describes the notion of a complex, or real,
1-un\-con\-di\-tio\-nal basic sequence $(\e_{rc})_{(r,c)\in I}$ of
elementary matrices in $\Sch^p$ (see Definition \ref{unc:def}.)

By a convexity argument, Question~\ref{q1} is
equivalent to the following question on Schur multiplication.

\begin{Q}\label{q2}
  Which matrix coefficients of an operator $x\in \Sch^p$ must vanish so
  that for all matrices $\varphi$ of complex, or real, numbers
  \begin{equation*}
    \|\varphi*x\|\le\sup|\varphi_{rc}|~\|x\|,
  \end{equation*}
  where $\varphi*x$ is
  the Schur (or Hadamard or entrywise) product defined by
  \begin{equation*}
    (\varphi*x)_{rc}=\varphi_{rc}x_{rc}?
  \end{equation*}
\end{Q}

In the case $p=\infty$, Grothendieck's inequality yields an estimation
for the norm of Schur multiplication by $\varphi$ in terms of the
projective tensor product $\ell_\Col^\infty\projtens\ell_\Row^\infty$:
this norm is equivalent to the supremum of the norm of those elements
of $\ell_\Col^\infty\projtens\ell_\Row^\infty$ whose
coefficient matrices are finite submatrices of $\varphi$. In the
framework of tensor algebras over discrete spaces, Question~\ref{q2}
turns out to describe as well the isometric counterpart to Varopoulos'
$\Varopoulos$-Sidon sets as well as to his sets of $\Varopoulos$-interpolation. The
following isometric question has however a different answer.

\begin{Q}\label{q3}
  Which coefficients of a tensor
  $u\in\ell_\Col^\infty\projtens\ell_\Row^\infty$ must vanish so that
  the norm of $u$ is the maximal modulus of its coefficients? 
\end{Q}

In our answer to Question~\ref{q2}, $\Sch^p$ and Schur multiplication
are treated as a noncommutative analogue to $\Ell^p$ and convolution.
The main step is a careful study of the Schatten-von-Neumann norm
$\|x\|=\bigl(\tr(x^*x)^{p/2}\bigr)^{1/p}$ for $p$ an even integer.
The rule of matrix multiplication provides an expression for this norm
as a series in the matrix coefficients of $x$ and their complex
conjugate, indexed by the $p$uples $(v_1,v_2,\dots,v_p)$ satisfying
$(v_{2i-1},v_{2i}),(v_{2i+1},v_{2i})\in I$, where $v_{p+1}=v_1$: see the computation in Eq.~\eqref{compu}. These are best understood
as closed walks of length $p$ on the bipartite graph $G$ canonically
associated to $I$: its vertex classes are $\Col$ and $\Row$ and its
edges are given by the couples in $I$. A structure theorem for closed
walks and a detailed study of the particular case in which $G$ is a cycle yield
the two following theorems that answer Questions~\ref{q1} and~\ref{q2}.

\begin{thm}\label{th4}
  Let $p\in(0,\infty]\setminus\{2,4,6,\dots\}$. If the
  sequence of elementary matrices $(\e_{rc})_{(r,c)\in I}$ is a real
  1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$, then the graph
  $G$ associated to $I$ contains no
  cycle. In this case, $I$ is even a set of $\Varopoulos$-interpolation with
  constant 1: every sequence $\varphi\in\ell^\infty_I$ may be
  interpolated by a tensor
  $u\in\ell_\Col^\infty\projtens\ell_\Row^\infty$ such that
  $\|u\|=\|\varphi\|$.
\end{thm}

\begin{thm}\label{th5}
  Let $p\in\{2,4,6,\dots\}$. The sequence
  $(\e_{rc})_{(r,c)\in I}$ is a complex, or real, 1-un\-con\-di\-tio\-nal
  basic sequence in $\Sch^p$ if and only if $G$ contains no cycle of
  length $4,6,\dots,p$.
\end{thm}

These theorems hold also for the complete counterparts to
1-un\-con\-di\-tio\-nal basic sequences in the sense of
Def.~\ref{unc:def}$(c)$.

In particular, if we denote by $\mathrm U_p$ the property that
$(\e_{rc})_{(r,c)\in I}$ is a 1-un\-con\-di\-tio\-nal basic sequence in
$\Sch^p$, then we obtain the following hierarchy:
\[\mathrm U_p\text{ for a $p\in(0,\infty]\setminus\{2,4,6,\dots\}$}\
\Rightarrow\ \cdots\ \Rightarrow\ \mathrm U_{2n+2}\ \Rightarrow\ 
\mathrm U_{2n}\ \Rightarrow\ \cdots\ \Rightarrow\ \mathrm U_2\ .\]

If $\Col$ and $\Row$ are finite, extremal graphs without cycles of
given lengths remain an ongoing area of research in graph theory.
Finite geometries seem to provide all known examples of such graphs
when $\Col$ and $\Row$ become large. Proposition~\ref{extr:prp} and
Remark~\ref{extr:rem} gather up known facts on this issue.

One may also avoid the terminology of graph theory and give an answer
in terms of polygons drawn in a matrix by joining matrix coordinates
with sides that follow alternately the row (horizontal) and the column
(vertical) direction of the matrix:

\begin{itemize}
\item Suppose that $p$ is not an even integer. If a pattern $I$
  contains the vertices of such a polygon, then there is an operator
  $x\in \Sch^p$ whose matrix coefficients vanish outside $I$ and
  whose norm depends on the sign of its matrix coefficients. This
  condition is also necessary.
\item If matrix coordinates of nonzero matrix coefficients of $x$ are
  the vertices of such a polygon with $n$ sides, then the norm of $x$ in
  $\Sch^p$ depends on the argument of its matrix coefficients for
  every even integer $p\ge n$; if the matrix coefficients of $x$ are real, then
  the norm of $x$ even depends on the sign of its matrix coefficients.
  These conditions are also necessary. 
\end{itemize}

An elementary example is given by the set
\begin{equation}
  I=\bigl\{(r,c)\in\Z/7\Z\times\Z/7\Z:r+c\in\{0,1,3\}\bigr\}.
\label{steiner}
\end{equation}
The associated bipartite graph is known as the Heawood graph (Fig.~\ref{fig:1}:) it is
the incidence graph of the Fano plane (the finite projective plane
$\mathrm{PG}(2,2)$,) which is the smallest generalised triangle, and
corresponds to the Steiner system $\mathrm S(2,3;7)$. It contains no
cycle of length 4, but every pair of vertices is contained in a
cycle of length 6. 
\begin{figure}[ht]
  \includegraphics{heawood}\centering
  \caption{The Heawood graph}
  \label{fig:1}
\end{figure}

Thus the $p$-trace norm of every matrix with
pattern
\[
\bordermatrix{
  &\scriptstyle0&\scriptstyle1&\scriptstyle2&\scriptstyle3&\scriptstyle4&\scriptstyle5&\scriptstyle6\cr
  \scriptstyle0&*&*&0&*&0&0&0\cr
  \scriptstyle1&*&0&*&0&0&0&*\cr
  \scriptstyle2&0&*&0&0&0&*&*\cr
  \scriptstyle3&*&0&0&0&*&*&0\cr
  \scriptstyle4&0&0&0&*&*&0&*\cr
  \scriptstyle5&0&0&*&*&0&*&0\cr
  \scriptstyle6&0&*&*&0&*&0&0\cr}
\]
does not depend on the sign of its coefficients
if and only if $p\in\{2,4\}$. 

These results give a complete description of the situation in which
$(\e_{rc})_{(r,c)\in I}$ is a 1-un\-con\-di\-tio\-nal basis of the space
$\Sch^p_I$ it spans in $\Sch^p$. If this is not the case, $\Sch^p_I$ might
still admit some other 1-un\-con\-di\-tio\-nal basis. This leads to the
following more general question. 

\begin{Q}\label{q4}
  For which sets $I$ does $\Sch^p_I$ admit some kind of almost
  1-un\-con\-di\-tio\-nal finite dimensional 
expansion of the identity?
\end{Q}
                                
The metric unconditional approximation property $(muap)$ provides a
formal definition for the object of Question~\ref{q4}: see
Def.~\ref{muap:def}. We obtain the following results.

\begin{thm}
  Let $p\in[1,\infty]\setminus\{2,4,6,\dots\}$. If $\Sch^p_I$ has
  real $(muap)$, then the distance of any two vertices that are not in
  the same vertex class is asymptotically infinite in $G$: their
  distance becomes arbitrarily large by deleting a finite number of
  edges from $G$.
\end{thm}

\begin{thm}
  Let $p\in\{2,4,6,\dots\}$. The space $\Sch^p_I$ has complex, or
  real, $(muap)$ if and only if any two vertices at distance $2j+1\le
  p/2$ are asymptotically at distance at least $p-2j+1$.
\end{thm}

We now turn to a detailed description of this article. In
Section~\ref{ss2}, we 
provide tools for the computation of Schur
multiplier norms. Section~\ref{ss3} characterises idempotent Schur
multipliers and $0,1$-tensors in $\ell_\Col^\infty\projtens\ell_\Row^\infty$
of norm 1. In Section~\ref{ss4}, we define the complex and real
unconditional constants of basic sequences of elementary matrices and
show that they are not equal in general. Section~\ref{V} looks back on
Varopoulos' results about tensor algebras over discrete spaces.
Section~\ref{ss6} puts the connection between $p$-trace norm and
closed walks of length $p$ in the concrete form of closed walk
relations. In Section~\ref{ss7}, we compute the norm of relative Schur
multipliers by signs in the case that $G$ is a cycle, and estimate the
corresponding unconditional constants. Section~\ref{ss8} is dedicated
to a proof of Th.~\ref{th4} and an answer to Question~\ref{q3}.
Section~\ref{ss9} establishes Th.~\ref{th5}. In Section~\ref{ss10},
we study the metric unconditional approximation property for spaces
$\Sch^p_I$. The final section provides four kinds of examples: sets
obtained by a transfer of $n$-independent subsets of a discrete abelian
group, Hankel sets, Steiner systems and Tits' generalised polygons. 

\begin{Ter}
  $\Col$ is the set of \emph{columns} and $\Row$ is the set of
  \emph{rows}, both finite or countable and if necessary indexed by
  natural numbers. $V$, the set of \emph{vertices}, is their
  disjoint union $\Col\amalg\Row$: if there is a risk of confusion, an
  element $n\in V$ that is a column (vs.~a row) will be
  referred to as ``col~$n$'' (vs.~``row~$n$''.) An
  \emph{edge} on $V$ is a pair $\{v,w\}\subseteq V$. A
  \emph{graph} on $V$ is given by a set of edges $E$. A
  \emph{bipartite} graph on $V$ with vertex classes $\Col$ and
  $\Row$ has only edges $\{r,c\}$ such that $c\in\Col$ and $r\in\Row$
  and may therefore be given alternatively by the set of couples
  $I=\bigl\{(r,c)\in\Row\times\Col:\{r,c\}\in E\bigr\}$: this will be
  our custom throughout the article. A bipartite graph on $V$ is
  \emph{complete} if its set of couples $I$ is the whole of $R\times
  C$. Two graphs are \emph{disjoint} if so are the sets of vertices
  of their edges. $I$ is a \emph{column section} if
  $(r,c),(r',c)\in I\Rightarrow r=r'$, and a \emph{row section} if $(r,c),(r,c')\in
  I\Rightarrow c=c'$.
  
  A \emph{walk} of length $s\ge0$ in a graph is a sequence
  $(v_0,\dots,v_s)$ of $s+1$ vertices such that
  $\{v_0,v_1\}$, $\ldots\,$,~$\{v_{s-1},v_s\}$ are edges of the graph. A walk
  is a \emph{path} if its vertices are pairwise distinct. The
  \emph{distance} of two vertices in a graph is the minimal length
  of a path in the graph that joins the two vertices; it is infinite
  if no such path exists. A \emph{closed} walk of length $p\ge0$ in
  a graph is a sequence $(v_1,\dots,v_p)$ of $p$ vertices such that
  $\{v_1,v_2\},\dots,\{v_{p-1},v_p\},\{v_p,v_1\}$ are edges of the
  graph. Note that $p$ is necessarily even if the graph is bipartite.
  A closed walk is a \emph{cycle} if its vertices are pairwise
  distinct. We take the convention that if a closed walk in a
  bipartite graph on $V=\Col\amalg\Row$ is nonempty, then its first
  vertex is a column vertex: $v_1\in\Col$. We shall identify a path
  and a cycle with its set of edges $\{r,c\}$ or the corresponding
  set of couples $(r,c)$.
  
  A bipartite graph on $V$ is a \emph{tree} if there is exactly one
  path between any two of its vertices. In this case, its vertices may
  be indexed by finite words over its set of vertices in the following
  way. Choose any row vertex $r$ as \emph{root} and index it by $\emptyset$. If $v$ is a vertex
  and $(r,c,\dots,v)$ is the unique path from
  $r$ to $v$, let the word $c\concat\cdots\concat v$ index $v$. Let
  $W$ be the set of all words thus formed. Then
  \begin{itemize}
  \item $\emptyset\in W$ and every beginning of a word in $W$ is also in
    $W$: if $w\in W\setminus\{\emptyset\}$, then $w$ is the
    concatenation $w'\concat v$ of a word $w'\in W$ with a letter $v$;
  \item words of even length index row vertices;
  \item words of odd length index column vertices;
  \item a pair of vertices is
    an edge exactly if their indices have the form $w$ and $w\concat
    v$, where $w$ is a word and $v$ is a letter.
  \end{itemize}
  A \emph{forest} is a union of pairwise disjoint trees; equivalently,
  it is a cycle free graph.
\end{Ter}

\begin{Not}
  Let $\T=\{z\in\C:|z|=1\}$. 
  
  The unit ball of a Banach space $X$ is denoted by $B_X$.

  Given an index set
  $I$ and $q\in I$, $\e_q$ is the sequence defined on $I$ as the
  indicator function $\chi_{\{q\}}$ of the singleton $\{q\}$.

  Let $I=\Row\times\Col$ and $q=(r,c)$. Then $\e_q=\e_{rc}$ 
  is the \emph{elementary matrix} identified with the operator from
  $\ell^2_\Col$ to $\ell^2_\Row$ that maps $\e_c$ on $\e_r$ and all
  other basis vectors on 0. The \emph{matrix coefficient} at
  coordinate $q$ of an operator $x$ from $\ell^2_\Col$ to $\ell^2_\Row$
  is $x_q=\tr\e_q^*x$ and its \emph{matrix representation} is
  $(x_q)_{q\in\Row\times\Col}=\sum_{q\in\Row\times
    \Col}x_q\e_q$. The \emph{support} of $x$ is $\{q\in\Row\times\Col:x_q\ne0\}$.
  
  The \emph{Schatten-von-Neumann class} $\Sch^p$, $0<p<\infty$,
  is the space of those compact operators $x$ from $\ell^2_\Col$ to
  $\ell^2_\Row$ such that
  $\|x\|_p^p=\tr|x|^p=\tr(x^*x)^{p/2}<\infty$.
  $\Sch^\infty$ is the space of compact operators with the
  operator norm. $\Sch^p$ is a quasi-normed space, and a Banach
  space if $p\ge1$. Let $(\Row_n\times\Col_n)_{n\ge0}$ be a sequence
  of finite sets that tends to $\Row\times\Col$. 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
  $\Sch^p$ if $p\ge1$. 
  
  For $I\subseteq\Row\times\Col$, the \emph{entry space}
  $\Sch^p_I$ is the subspace of those $x\in \Sch^p$ whose
  support is a subset of $I$. $\Sch^p_I$ is also the closed
  subspace of $\Sch^p$ spanned by $(\e_q)_{q\in I}$.
  
  The $\Sch^p$-valued Schatten-von-Neumann class
  $\Sch^p(\Sch^p)$ is the space of those compact operators
  $x$ from $\ell^2_\Col$ to $\ell^2_\Row(\Sch^p)$ such that
  $\|x\|_p^p=\tr(\tr|x|^p)<\infty$, where the inner trace is the
  $\Sch^p$-valued analogue of the usual trace: such operators
  have an $\Sch^p$-valued matrix representation and their support
  is defined as in the scalar case. An element
  $x\in\Sch^p(\Sch^p)$ can also be considered as a compact
  operator from $\ell^2_\Col(\ell_2)=\ell_2\otimes_2\ell^2_\Col$ to
  $\ell^2_\Row(\ell_2)=\ell_2\otimes_2\ell^2_\Row$ such that
  $\|x\|_p^p=\tr\otimes\tr|x|^p<\infty$; the matrix coefficient
  of $x$ at $q$ is then $x_q = (\Id_{\Sch^p}\otimes\tr)
  \bigl((\Id_{\ell_2}\otimes\e_q^*)x\bigr)$ and its matrix
  representation is $\sum_{q\in\Row\times\Col}x_q\otimes\e_q$. The
  entry space $\Sch^p_I(\Sch^p)$ is defined in the same way
  as $\Sch^p_I$.
  
  A \emph{relative Schur multiplier} on $\Sch^p_I$ is a
  sequence $\varphi=(\varphi_q)_{q\in I}\in\C^I$ such that the
  associated \emph{Schur multiplication operator} $\Schur_\varphi$
  defined by $\e_q\mapsto\varphi_q\e_q$ for $q\in I$ is bounded on
  $\Sch^p_I$. The Schur multiplier $\varphi$ is furthermore
  \emph{completely bounded} (c.b.~for short) on $\Sch^p_I$ if
  $\Id_{\Sch^p}\otimes\Schur_\varphi$, the operator defined by
  $x_q\e_q\mapsto\varphi_qx_q\e_q$ for $x_q\in \Sch^p$ and $q\in
  I$, is bounded on $\Sch^p_I(\Sch^p)$ (see
  \cite[Lemma~1.7]{pi98}.) The norm of $\varphi$ is the norm of
  $\Schur_\varphi$ and its \emph{complete norm} is the norm of
  $\Id_{\Sch^p}\otimes\Schur_\varphi$.  This norm is the supremum
  of the norm of its restrictions to finite rectangle sets
  $\Row'\times\Col'$. Note that $\varphi$ is a Schur multiplier
  on $\Sch^\infty$ if and only if, for every bounded operator
  $x\colon\ell^2_\Col\to\ell^2_\Row$, $(\varphi_qx_q)$ is the matrix representation
  of a bounded operator; also 
  $\varphi$ is automatically c.b.~on $\Sch^\infty$
  \cite[Th.~5.1]{pi01}. We used \cite{pi98,pi01} as a reference.
  
  Let $G$ be a compact abelian group endowed with its normalised Haar
  measure. Let $\Gamma=\hat G$ be the dual group of characters on $G$.
  The \emph{spectrum} of an integrable function $f$ on $G$ is
  $\{\gamma\in\Gamma:\hat f(\gamma)\ne0\}$. Let
  $\Lambda\subseteq\Gamma$. If $X$ is a space of integrable functions
  on $G$, then $X_\Lambda$ is the \emph{translation invariant}
  subspace of those $f\in X$ whose spectrum is a subset of $\Lambda$.
  
  Let $X$ be the space of continuous functions $\Cont(G)$ or the
  Lebesgue space $\Ell^p(G)$ with $0<p<\infty$. Then $X_\Lambda$ is
  also the closed subspace of $X$ spanned by $\Lambda$.
  A \emph{relative Fourier multiplier} on $X_\Lambda$ is a sequence
  $\mu=(\mu_\gamma)_{\gamma\in\Lambda}\in\C^\Lambda$ such that the
  associated convolution operator $\Fourier_\mu$ defined by
  $\gamma\mapsto\mu_\gamma\gamma$ for $\gamma\in\Lambda$ is bounded on
  $X_\Lambda$. The Fourier multiplier $\mu$ is furthermore c.b.\ if
  $\Id_{\Sch^p}\otimes\Fourier_\mu$, the operator defined by
  $a_\gamma\gamma\mapsto \mu_\gamma a_\gamma\gamma$ for
  $a_\gamma\in\Sch^p$ and $\gamma\in\Lambda$, is bounded on the
  $\Sch^p$-valued space $X_\Lambda(\Sch^p)$ (where
  $p=\infty$ if $X=\Cont(G)$.) The norm of
  $\mu$ is the norm of $\Fourier_\mu$ and its complete norm is the
  norm of $\Id_{\Sch^p}\otimes\Fourier_\mu$. Note that $\mu$ is a
  Fourier multiplier on $\Cont_\Lambda(G)$ if and only if, for
  every $f\in\Ell_\Lambda^\infty(G)$, $\sum\mu_\gamma\hat
  f(\gamma)\gamma$ is the Fourier series of an element of
  $\Ell_\Lambda^\infty(G)$: $\mu$ is a
  \emph{relative Fourier multiplier on $\Ell^\infty(G)$}; also $\mu$ is automatically c.b.~on
  $\Cont_\Lambda(G)$ \cite[Cor.\ 3.18]{pi01}.

  Let $X,Y$ be Banach spaces and $u\in X\otimes Y$. Its \emph{projective}
  tensor norm is
  \begin{equation*}
    \|u\|_{X\projtens Y}=\inf\biggl\{\sum_{j=1}^n\|x_j\|\,\|y_j\|:u=\sum_{j=1}^nx_j\otimes
    y_j\biggr\}
  \end{equation*}
  and $X\projtens Y$ is the completion of $X\otimes Y$ with respect to
  this norm. Note that $\ell_\infty^n\projtens \ell_\infty^m\subset
  \cnaught\projtens\cnaught$ because $\ell_\infty^n$ and
  $\ell_\infty^m$ are 1-complemented in $\cnaught$, and that
  $\cnaught\projtens\cnaught\subset\ell_\infty\projtens\ell_\infty$
  because $\ell_\infty$ is the bidual of $\cnaught$. 
  
  Let $\sum x_j\otimes y_j$ be any representation of the tensor $u$.
  If $\xi\otimes\eta\in X^*\otimes Y^*$, we define
  $\langle\xi\otimes\eta,u\rangle=\sum\langle\xi,x_j\rangle\langle\eta,y_j\rangle$.
  The \emph{injective} tensor norm of $u$ is
  \begin{equation*}
    \|u\|_{X\injtens Y} =
    \sup_{(\xi,\eta)\in B_{X^*}\times B_{Y^*}}|\langle\xi\otimes\eta,u\rangle|
  \end{equation*}
  and $X\injtens Y$ is the completion of $X\otimes Y$ with
  respect to this norm. 

  If $X$ and $Y$ are both finite dimensional, then
  \begin{equation*}
    \bigl(X\injtens Y\bigr)^*=X^*\projtens Y^*\quad\text{and}\quad
    \bigl(X\projtens Y\bigr)^*=X^*\injtens Y^*.
  \end{equation*}
  Further
  $\bigl(\cnaught\projtens\cnaught\bigr)^*=\ell_1\injtens\ell_1$: in
  fact, $\bigl(\cnaught\projtens\cnaught\bigr)^*$ may be identified
  with the space of bounded operators from $\cnaught$ to $\ell_1$ and
  $\ell_1\injtens\ell_1$ may be identified with the closure of finite
  rank operators in that space, and they are the same because every
  bounded operator from $\cnaught$ to $\ell_1$ is compact and $\ell_1$
  has the approximation property. 

  If $X$ is a sequence space on
  $\Col$ and $Y$ is a sequence space on $\Row$, then the
  \emph{coefficient} of the tensor $u$ at $(r,c)$ is
  $\langle\e_c\otimes\e_r,u\rangle$. Its \emph{support} is the set
  of coordinates $(r,c)$ of its nonvanishing coefficients. One may use
  \cite{ry02} as a reference.
  
\end{Not}

\section{Relative Schur multipliers}\label{ss2}

\noindent The following proposition is a straightforward consequence of
\cite{pps89}.

\begin{prp}\label{pinf}
  Let $I\subseteq\Row\times\Col$ and $\varphi$ be a Schur multiplier
  on $\Sch^\infty_I$ with norm $D$. Then $\varphi$ is also a c.b.~Schur multiplier
  on $\Sch^p_I$ for every $p\in(0,\infty]$, with complete norm bounded by $D$.
\end{prp}

\begin{proof}
  We may assume that $D=1$. Let $\Row'\times\Col'$ be any finite subset of
  $\Row\times\Col$. By \cite[Th.~3.2]{pps89}, there exist vectors
  $w_c$ and $v_r$ of norm at most 1 in a Hilbert space
  $H$ such that $\varphi_{rc}=\langle w_c,v_r\rangle$ for every
  $(r,c)\in I\cap\Row'\times\Col'$. If we define $W\colon\ell^2_{\Col'}\to\ell^2_{\Col'}(H)$ and
  $V\colon\ell^2_{\Row'}\to\ell^2_{\Row'}(H)$ by
  $W\zeta=(\zeta_cw_c)_{c\in\Col'}$ and $V\eta=(\eta_rv_r)_{r\in\Row'}$,
  then $V$ and $W$ have norm at most 1, and the proposition follows from the factorisation
  \begin{equation*}
    \Schur_\varphi x=V^*(x\otimes\Id_H)W
  \end{equation*}
  for every $x$ with support in $I\cap\Row'\times\Col'$.
\end{proof}

\begin{rem}
  \'Eric Ricard showed us an elementary proof
  that a Schur multiplier on $\Sch^\infty_I$ is automatically
  c.b., included here by his kind permission. A Schur multiplier
  $\varphi$ is bounded on $\Sch^\infty_I$ by a constant $D$ if and only if
  \begin{equation}\label{b}
    \forall\,\xi\in
    B_{\Sch^\infty_I}~\forall\,\eta\in B_{\ell^2_\Row}~\forall\,\zeta\in
    B_{\ell^2_\Col}\quad\biggl|\sum_{(r,c)\in
      I}\eta_r\varphi_{rc}\xi_{rc}\zeta_c\biggr|\le D.
  \end{equation}
  It is furthermore completely bounded on $\Sch^\infty_I$ by $D$ if
  \begin{equation}\label{cb}
    \forall\,x\in
    B_{\Sch^\infty_I(\Sch^\infty)}~\forall\,y\in B_{\ell^2_\Row(\ell_2)}~\forall\,z\in
    B_{\ell^2_\Col(\ell_2)}\quad\biggl|\sum_{(r,c)\in I}\varphi_{rc}\langle
    y_r,x_{rc}z_c\rangle\biggr|\le D.
  \end{equation}
  Suppose that $x,y,z$ are as quantified in Ineq.~\eqref{cb}. Let
  \[
  \xi_{rc}=\langle y_r/\|y_r\|,x_{rc}z_c/\|z_c\|\rangle ,~
  \eta_r=\|y_r\|_{\ell_2} ~\text{and}~ \zeta_c=\|z_c\|_{\ell_2}.
  \] 
  Then
  $\|\eta\|_{\ell^2_\Row},\|\zeta\|_{\ell^2_\Col}\le1$ and
  \begin{multline*}
    \|\xi\|=\sup\biggl\{\biggl| \sum_{(r,c)\in I} \langle \alpha_ry_r/\|y_r\|_{\ell_2},
    x_{rc}\beta_c z_c/\|z_c\|_{\ell_2} \rangle \biggr| : \alpha\in B_{\ell^2_\Row},\,
    \beta\in B_{\ell^2_\Col} \biggr\}\\
    \le \|x\| 
    \sup_{\alpha\in B_{\ell^2_\Row}}
    \bigl\|\bigl(\alpha_ry_r/\|y_r\|_{\ell_2}\bigr)\bigr\|_{\ell^2_\Row(\ell_2)}
    \sup_{\beta\in B_{\ell^2_\Col}}
    \bigl\|\bigl(\beta_cz_c/\|z_c\|_{\ell_2}\bigr)\bigr\|_{\ell^2_\Col(\ell_2)}\le 1, 
  \end{multline*}
  so that Ineq.~\eqref{b} implies Ineq.~\eqref{cb}.
\end{rem}

The fact that the canonical basis of an $\ell^2$ space is
1-un\-con\-di\-tio\-nal yields that Schatten-von-Neumann norms are
\emph{matrix unconditional} in the terminology of \cite{sc78}:
\begin{equation}
  \label{mu}
  \forall\,\zeta\in\T^\Col~\forall\,\eta\in\T^\Row\quad
  \Bigl\|\sum_{(r,c)\in\Row\times\Col}\zeta_c\eta_ra_{rc}\e_{rc}\Bigr\|_p=
  \Bigl\|\sum_{(r,c)\in\Row\times\Col}a_{rc}\e_{rc}\Bigr\|_p
\end{equation}
for every finitely supported sequence of complex or
$\Sch^p$-valued coefficients $a_{rc}$. Let $\zeta\otimes\eta$
denote the \emph{elementary} Schur multiplier
$(\zeta_c\eta_r)_{(r,c)\in\Row\times\Col}$. Equation~\eqref{mu} shows that if
$\zeta\in\T^{\Col}$ and $\eta\in\T^{\Row}$, then
$\Schur_{\zeta\otimes\eta}$ is an isometry on every $S^p$. This yields
that if 
$\zeta\in\ell^\infty_\Col$, $\eta\in\ell^\infty_\Row$, then the
complete norm of $\Schur_{\zeta\otimes\eta}$ is 
$\|\zeta\|_{\ell^\infty_\Col}\|\eta\|_{\ell^\infty_\Row}$ on every
$\Sch^p$. 

Relative Schur multipliers also have a central place among operators
on $\Sch^p_I$ because they appear as the range of a contractive
projection defined by the following averaging scheme.

\begin{dfn}
  Let $T\colon\Sch^p_J\to\Sch^p_I$ be an operator. Let
  $\Row'\times\Col'$ be a finite subset of $\Row\times\Col$ and let
  $P_{\Row'\times\Col'}$ be the contractive projection onto $\Sch^p_{\Row'\times\Col'}$
  defined by the Schur multiplier
  $\chi_{\Col'}\otimes\chi_{\Row'}$. Then the \emph{average of $T$
    with respect to $\Row'\times\Col'$} is given by
  \begin{equation}\label{ave}
    [T]_{\Row'\times\Col'}(x) = \int_{\T^{\Row}}\mathrm{d}\eta\,
    \int_{\T^{\Col}}\mathrm{d}\zeta\,
    \Schur_{\zeta^*\otimes\eta^*}
    P_{\Row'\times\Col'}T(\Schur_{\zeta\otimes\eta}x),
  \end{equation}
  where $\zeta^*=(\overline{\zeta_c})_{c\in\Col}$ and
  $\eta^*=(\overline{\eta_r})_{r\in\Row}$. 
\end{dfn}

\begin{prp}\label{aver}
  Let $T\colon\Sch^p_J\to\Sch^p_I$ be an operator and
  $\Row'\times\Col'$ a finite subset of $\Row\times\Col$. Then
  $[T]_{\Row'\times\Col'}$ is a Schur multiplication operator from
  $\Sch^p_J$ to $\Sch^p_{I\cap\Row'\times\Col'}$ such that
  $\|[T]_{\Row'\times\Col'}\|\le\|T\|$. In fact,
  $[T]_{\Row'\times\Col'}=M_{\varphi^{\Row'\times\Col'}}$ with 
  \begin{equation*}
    \varphi^{\Row'\times\Col'}_{rc}=
    \begin{cases}
      \tr\e_{rc}^*T(\e_{rc})&\text{if
        $(r,c)\in J\cap\Row'\times\Col'$}\\
      0&\text{if $(r,c)\in J\setminus\Row'\times\Col'$.}
    \end{cases}
  \end{equation*}
  If $T$ is a projection onto $\Sch^p_I$, then
  $\varphi^{\Row'\times\Col'}=\chi_{I\cap\Row'\times\Col'}$, so that
  $[T]_{\Row'\times\Col'}$ is a projection onto
  $\Sch^p_{I\cap\Row'\times\Col'}$. Let
  $\varphi=\bigl(\tr\e_q^*T(\e_q)\bigr)_{q\in J}$. Then
  $\|\Schur_\varphi\|\le\|T\|$ and we define the \emph{average of $T$} by $[T]=\Schur_\varphi$.  
\end{prp}

\begin{proof}
  Formula~\eqref{ave} shows that
  $\|[T]_{\Row'\times\Col'}(x)\|\le\|T\|\,\|x\|$. We have
  \begin{align*}
    [T]_{\Row'\times\Col'}(\e_{rc}) &=
    \int_{\T^{\Row}}\mathrm{d}\eta\, \int_{\T^{\Col}}\mathrm{d}\zeta\,
    \Schur_{\zeta^*\otimes\eta^*}
    P_{\Row'\times\Col'}T(\zeta_c\eta_r\e_{rc})\\
    &= \int_{\T^{\Row}}\mathrm{d}\eta\,
    \int_{\T^{\Col}}\mathrm{d}\zeta\, \zeta_c\eta_r
    \Schur_{\zeta^*\otimes\eta^*} \sum_{(r',c')\in\Row'\times\Col'}
    \tr\bigl(\e_{r'c'}^*T(\e_{rc})\bigr) \e_{r'c'}\\
    &= \sum_{(r',c')\in\Row'\times\Col'}
    \int_{\T^{\Row}}\mathrm{d}\eta\, \int_{\T^{\Col}}\mathrm{d}\zeta\,
    \zeta_c\eta_r \tr\bigl(\e_{r'c'}^*T(\e_{rc})\bigr)
    \zeta_{c'}^{-1}\eta_{r'}^{-1}\e_{r'c'}
    = \varphi^{\Row'\times\Col'}_{rc}\e_{rc}.
  \end{align*}
  As the norm of a Schur multiplier is the supremum of the norm of its
  restrictions to finite rectangle sets, this shows that
  $\varphi$ is a Schur multiplier on $\Sch^p_J$ and
  $\|\Schur_\varphi\|\le\|T\|$. 
  If $T$ is a projection onto $\Sch^p_I$, note that
  $\tr\e_{rc}^*T(\e_{rc})=\chi_I(r,c)$.
\end{proof}

The following proposition relates Fourier multipliers to Herz-Schur
multipliers in the fashion of \cite[Th.~6.4]{pi01} and will be very useful
in the exact computation of the norm of certain relative Schur multipliers.

\begin{prp}\label{transfer}
  Let $\Gamma$ be a countable discrete abelian group and
  $\Lambda\subseteq\Gamma$. Let $\Row$ and $\Col$ be two copies of
  $\Gamma$ and consider $I=\{(r,c)\in\Row\times\Col:r-c\in\Lambda\}$.
  Let $\varphi\in\C^I$ such that there is $\mu\in\C^\Lambda$ with
  $\varphi(r,c)=\mu(r-c)$ for all $(r,c)\in I$. Let $G=\hat\Gamma$, so
  that $\Gamma$ is the group of characters on the compact group
  $G$. Let $p\in(0,\infty]$. 
  \begin{enumerate}
  \item The complete norm of the relative Schur multiplier $\varphi$
    on $\Sch^p_I$ is bounded by the complete norm of the relative
    Fourier multiplier $\mu$ on
    $\Ell^p_\Lambda(G)$. 
  \item Suppose that $\Gamma$ is finite. The norm of the relative Fourier multiplier $\mu$ on
    $\Ell^p_\Lambda(G)$ is bounded by the norm of the relative Schur
    multiplier $\varphi$ on $\Sch^p_I$. The same holds for
    complete norms.
  \end{enumerate}
\end{prp}

\begin{rem}
  Part~$(b)$ is just an abstract counterpart to \cite[Chapter~6,
  Lem\-ma~3.8]{pe03}, where the case of the finite cyclic group
  $\Gamma=\Z/n\Z$ is treated.
\end{rem}

\begin{proof}$(a)$ is \cite[Lemma 8.1.4]{pi98}:
   for all $a_q\in \Sch^p$, of which only a finite number
   are nonzero, and all $g\in G$, we have by matrix unconditionality
   (Eq.~\eqref{mu})
   \begin{multline}\label{tr1}
     \Bigl\|\sum_{q\in I}a_q\e_q\Bigr\|_{\Sch^p_I(\Sch^p)} = \Bigl\|\sum_{(r,c)\in
       I}r(g)c(g)^{-1}a_{rc}\e_{rc}\Bigr\|_{\Sch^p_I(\Sch^p)}\\
     = \biggl\| \sum_{\gamma\in\Lambda}
     \biggl(\sum_{\substack{(r,c)\in I\\r-c=\gamma}}
     a_{rc}\e_{rc}\biggr)\gamma(g)\biggr\|_{\Sch^p_I(\Sch^p)} =
     \biggl\|\sum_{\gamma\in\Lambda} \biggl(\sum_{\substack{(r,c)\in
         I\\r-c=\gamma}}
     a_{rc}\e_{rc}\biggr)\gamma\biggr\|_{\Ell^p_\Lambda(G,\Sch^p(\Sch^p))}.
   \end{multline}
   This yields an isometric embedding of $\Sch^p_I(\Sch^p)$
   in $\Ell^p_\Lambda(G,\Sch^p_I(\Sch^p))$. As
   $\Sch^p(\Sch^p)$ may be identified with 
   $\Sch^p(\ell^2_\Gamma(\ell^2))$,
   \begin{equation*}\label{tr2}
     \Bigl\| \sum_{q\in I} \varphi_qa_q\e_q
     \Bigr\|_{\Sch^p_I(\Sch^p)} = \biggl\|
     \sum_{\gamma\in\Lambda} \mu_\gamma \biggl(
     \sum_{\substack{(r,c)\in I\\r-c=\gamma}} a_{rc}\e_{rc} \biggr)
     \gamma \biggr\|_{\Ell^p_\Lambda(G,\Sch^p(\Sch^p))}
     \le \| \Id\otimes\Fourier_\mu \| \Bigl\| \sum_{q\in I} a_q\e_q
     \Bigr\|_{\Sch^p_I(\Sch^p)}.
   \end{equation*}

$(b)$.  Let us embed $\Ell^p_\Lambda(G)$ into $\Sch^p_I$ by
$f\mapsto\fourier_{\hat f}$, where
$\fourier_{\hat f}\colon\ell^2_\Col\to\ell^2_\Row$ is the convolution operator defined by 
\[\fourier_{\hat f}\e_c
=\hat f*\e_c
=\sum_{\gamma\in\Lambda}\hat f(\gamma)\e_\gamma*\e_c
=\sum_{r-c\in\Lambda}\hat f(r-c)\e_r:\]
$\fourier_{\hat f}$ has the matrix representation $\sum_{(r,c)\in I}\hat
f(r-c)\e_{rc}$. The characters $g\in G$ form an 
orthonormal basis for $\ell^2_\Col$ such that $\fourier_{\hat f}g=f(g)g$: 
therefore 
\[{\|\fourier_{\hat f}\|}_p=\Bigr(\sum_{g\in
  G}|f(g)|^p\Bigr)^{1/p}=(\card G)^{1/p}{\|f\|}_{\Ell^p(G)}.\] As
$\Schur_{\varphi}\fourier_{\hat f}=\fourier_{\widehat{\Fourier_\mu f}}$, this shows that
the norm of $\mu$ on $\Ell^p_\Lambda(G)$ is the norm of $\varphi$ on
the subspace of circulant matrices in $\Sch^p_I$. The same
holds for complete norms.
\end{proof}

\section{Idempotent Schur multipliers of norm 1}\label{ss3}

\noindent A Schur multiplier is \emph{idempotent} if it is the indicator
function $\chi_I$ of some set $I\subseteq\Row\times\Col$; if $\chi_I$
is a Schur multiplier on $\Sch^p$, then it is a projection of $\Sch^p$ onto
$\Sch^p_I$.  Idempotent Schur multipliers on $\Sch^p$ and tensors in
$\ell^\infty_\Col\projtens \ell^\infty_\Row$ with $0,1$ coefficients
of norm 1 may be characterised by the combinatorics of $I$.

\begin{prp}\label{complem}
  Let $I\subseteq\Row\times\Col$ be nonempty and $0<p\ne2<\infty$. The following
  are equivalent. 
  
  \begin{enumerate}
  \item For every finite rectangle set $\Row'\times\Col'$
    intersecting $I$
    \begin{equation*}
      \biggl\| \sum_{(r,c)\in I\cap\Row'\times\Col'} \e_c\otimes\e_r
      \biggr\| _{\ell^\infty_\Col\projtens \ell^\infty_\Row}=1.
    \end{equation*}
  \item $\Sch^p_I$ is completely 1-complemented in $\Sch^p$.
  \item $\Sch^p_I$ is 1-complemented in $\Sch^p$.
  \item $I$ is a union of pairwise disjoint complete bipartite graphs: there
    are pairwise disjoint sets $\Row_j\subseteq\Row$ and pairwise disjoint sets
    $\Col_j\subseteq\Col$ such that $I=\bigcup\Row_j\times\Col_j$.
  \end{enumerate}
  
\end{prp}

Property $(d)$ means that the pattern $I$ is, up to a permutation of columns and rows, block-diagonal:
\[
\bordermatrix{
  &\scriptstyle C_1&\scriptstyle C_2&\scriptstyle C_3&\scriptstyle\cdots\cr
  \scriptstyle R_1&*&0&0&\cdots\cr
  \scriptstyle R_2&0&*&0&\ddots\cr
  \scriptstyle R_3&0&0&*&\ddots\cr
  \scriptstyle\hfil\vdots&\vdots&\ddots&\ddots&\ddots\cr
  }\text.
\]
 

\begin{proof}
  $(b)\Rightarrow(c)$ is trivial.

  $(a)\Rightarrow(b)$.  The complete norm of a Schur multiplier $\varphi$ on
  $\Sch^p$ is the supremum of the complete norm of its restrictions
  $\varphi'=(\varphi_q)_{q\in\Row'\times\Col'}$ to finite rectangle
  sets $\Row'\times\Col'$. Furthermore, the complete norm of an elementary
  Schur multiplier $(\eta_c\zeta_r)_{(r,c)\in\Row\times\Col}=\eta\otimes\zeta$ on
  $\Sch^p$ equals
  $\|\eta\|_{\ell^\infty_\Col}\|\zeta\|_{\ell^\infty_\Row}$.
  
  $(c)\Rightarrow(d)$. If $I$ is not a union of pairwise disjoint
  complete bipartite graphs, then there are $r_0,r_1\in\Row$ and 
  $c_0,c_1\in\Col$ such that 
  \begin{equation*}
    I'=I\cap\{r_0,r_1\}\times\{c_0,c_1\}=\{(r_0,c_0),(r_1,c_0),(r_0,c_1)\}.
  \end{equation*}
  By Proposition~\ref{aver}, the average of a contractive projection of $\Sch^p$ onto
  $\Sch^p_I$ with respect to $\{r_0,r_1\}\times\{c_0,c_1\}$ would
  be the contractive projection associated to the Schur multiplier
  $\chi_{I'}$.  Let $x(t)$, $t\in\R$, be the
  operator from $\ell^2_C$ to $\ell^2_R$ whose matrix coefficients
  vanish except for its $\{r_0,r_1\}\times\{c_0,c_1\}$ submatrix,
  which is
  $\begin{pmatrix}1&\sqrt2\\
    \sqrt2&t\end{pmatrix}$. Its eigenvalues are
  \begin{equation*}
    \frac{1+t+\sqrt{9-2t+t^2}}2=2+\frac t3+o(t),~
    \frac{1+t-\sqrt{9-2t+t^2}}2=-1+\frac{2t}3+o(t),
  \end{equation*}
  so that
  \begin{equation*}
    \left\{
      \begin{aligned}
        &\|x(t)\|_\infty=2+t/3+o(t)\\
        &\|x(t)\|_p^p=2^p+1+p(2^p-4)t/6+o(t)\quad\text{for}~0<p<\infty
      \end{aligned}
    \right.
  \end{equation*}
  and therefore $\|\chi_{I'}*x(t)\|_p=\|x(0)\|_p>\|x(t)\|_p$ for some $t\ne0$ if
  $p\ne2$.
  
  $(d)\Rightarrow(a)$. Suppose $(d)$ and let $\Row'\times\Col'$
  intersect $I$. Then there are pairwise disjoint sets $\Row'_j$ and
  pairwise disjoint sets $\Col'_j$ such that $I\cap\Row'\times
  \Col'=\Row'_1\times\Col'_1\cup\dots\cup\Row'_n\times\Col'_n$ and
  \begin{equation*}
    \sum_{(r,c)\in I\cap\Row'\times\Col'}\e_c\otimes\e_r = 
    \sum_{j=1}^{n}\chi_{\Col'_j}\otimes\chi_{\Row'_j} = 
    \average_{\epsilon_j=\pm1}
    \biggl(\sum_{j=1}^{n}\epsilon_j\chi_{\Col'_j}\biggr)
    \otimes\biggl(\sum_{j=1}^{n}\epsilon_j\chi_{\Row'_j}\biggr)
  \end{equation*}
  which is an average of elementary tensors of norm 1, so that its
  projective tensor norm is bounded by 1, and actually is equal to
  1. 
\end{proof}

\begin{rem}
  Note that the proof of Prop.~\ref{complem} shows that the norm of a projection
  $\Schur_{\chi_I}\colon \Sch^\infty\to \Sch^\infty_I$ is either 1 or at least $2/\sqrt3$,
  as
  \begin{equation*}
    \left\|\begin{pmatrix}1&\sqrt2\\ \sqrt2&-1\end{pmatrix}
    \right\|_\infty=\sqrt3,\qquad
    \left\|\begin{pmatrix}1&\sqrt2\\ \sqrt2&0\end{pmatrix}\right\|_\infty=2.
  \end{equation*}
  This is a noncommutative analogue to the fact that an idempotent measure on a
  locally compact abelian group $G$ has either norm 1 or at least
  $\sqrt5/2$ \cite[Th.~3.7.2]{ru62}. The norm of $\Schur_{\chi_I}$ actually equals
  $2/\sqrt3$ for $I=\{(0,0),(0,1),(1,0)\}$, as shown in \cite[Lemma
  3]{liv95}. In fact, the following decomposition holds:
  \begin{multline*}
    \e_0\otimes\e_0+\e_0\otimes\e_1+\e_1\otimes\e_0=\\
    \bigl(
    (\e^{-\iu\pi/12},\e^{\iu\pi/4})\otimes
    (\e^{-\iu\pi/12},\e^{\iu\pi/4})+
    (\e^{\iu\pi/12},\e^{-\iu\pi/4})\otimes
    (\e^{\iu\pi/12},\e^{-\iu\pi/4})\bigr)/\sqrt3.
  \end{multline*}
\end{rem}

\begin{rem}
  Results related to the equivalence of $(c)$ with $(d)$ have been obtained
  independently by Banks and Harcharras \cite{bh04}.
\end{rem}

\section{\texorpdfstring{Unconditional basic sequences in $\Sch^p$}{Unconditional basic sequences of elementary matrices in Schatten classes}}

\label{ss4}

\begin{dfn}
  \label{unc:def}
  Let $0<p\le\infty$ and $I\subseteq\Row\times\Col$. Let $\Sign=\T$
  (vs.~$\Sign=\{-1,1\}$.) 
  
  \begin{enumerate}
  \item $I$ is an \emph{unconditional basic sequence} in $\Sch^p$ if there is a
    constant $D$ such that
    \begin{equation}\label{u}
      \Bigl\|\sum_{q\in I}\epsilon_qa_q\e_q\Bigr\|_p \le D
      \Bigl\|\sum_{q\in I} a_q\e_q\Bigr\|_p
    \end{equation}
    for every  choice of signs $\epsilon_q\in\Sign$
    and every finitely supported sequence of complex coefficients $a_q$. Its
    \emph{complex \klammer{vs.~real} unconditional constant} is the
    least such constant $D$. 
  \item $I$ is a \emph{completely} unconditional basic sequence in $\Sch^p$
    if there is a constant $D$ such that \eqref{u} holds for every choice of signs
    $\epsilon_q\in\Sign$ and every finitely supported sequence of operator
    coefficients $a_q\in \Sch^p$. Its complex (vs.~real) \emph{complete}
    unconditional constant is the least such constant $D$.
  \item $I$ is a complex
    (vs.~real, complex completely, real completely)
    \emph{1-un\-con\-di\-tio\-nal} basic sequence in $\Sch^p$ if its complex
    (vs.~real, complex complete, real complete) unconditional
    constant is 1: Inequality \eqref{u} turns into the equality
    \begin{equation*}
      \Bigl\|\sum_{q\in I}\epsilon_qa_q\e_q\Bigr\|_p =
      \Bigl\|\sum_{q\in I} a_q\e_q\Bigr\|_p.
    \end{equation*}
  \end{enumerate}
  
\end{dfn}

If Inequality~\eqref{u} holds for every real choice of signs, then it
also holds for every complex choice of signs at the cost of replacing
$D$ by $D\pi/2$ (see \cite{se97},) so that there is no need to
distinguish between complex and real unconditional basic sequences.

The notions defined in $(a)$ and $(b)$ are called \emph{$\sigma(p)$ sets} and
\emph{complete $\sigma(p)$ sets} in \cite[\S4]{ha98} and \cite{hno01} (see
also the survey \cite[\S\,9]{px03}.) The notions defined in $(c)$ are
their isometric counterparts.

By \cite[proof of Cor.~4]{sc78}, the real unconditional constant of
any basis of $\Sch^p_I$ cannot be lower than a fourth of the real
unconditional constant of $I$ in $\Sch^p$.

\begin{exa}
  A single column $\Row\times\{c\}$, a single row $\{r\}\times\Col$,
  the diagonal set
  ${\{(\text{row}\,n,\allowbreak\text{col}\,n)\}}_{n\in\N}$ if $\Row$
  and $\Col$ are copies of $\N$, are 1-un\-con\-di\-tio\-nal basic sequences
  in all $\Sch^p$. In fact, every column section and every row section
  (this is the terminology of \cite[Def.~4.3]{va69}) is a
  1-un\-con\-di\-tio\-nal basic sequence; note that the length of every path
  in the corresponding graph is at most 2.
\end{exa}

Note that the set $I$ is a (completely) 1-un\-con\-di\-tio\-nal basic
sequence in $\Sch^p$ if and only if the relative Schur multipliers
by signs on $\Sch^p_I$ define (complete) isometries. This yields by
Prop.~\ref{pinf}:

\begin{prp}
  \label{pi}
  Let $I\subseteq\Row\times\Col$ and $0<p\le\infty$. If $I$ is
  a real \klammer{vs.~complex} 1-un\-con\-di\-tio\-nal basic sequence in
  $\Sch^\infty$, then $I$ is also a real \klammer{vs.~complex} completely
  1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$.
\end{prp}

\begin{exa}
  If $\Row=\Col=\{0,\dots,n-1\}$, $1\le p\le\infty$ and $I=\Row\times
  \Col$, then the complex unconditional constant of the basis of
  elementary matrices in $\Sch^p$ is $n^{|1/2-1/p|}$ and coincides with
  its complete unconditional constant (see \cite[Lemma
  8.1.5]{pi98}.) This is also the real unconditional constant if
  $n=2^k$ is a power of 2 as the norm of Schur
  multiplication by the $k$th tensor power $
  \begin{pmatrix}
    1&1\\
    1&-1
  \end{pmatrix}
  ^{\otimes k}$ (the $k$th Walsh matrix) on $\Sch^p$ is
  $\bigl(2^{|1/2-1/p|}\bigr)^k=n^{|1/2-1/p|}$. Let us now show that if
  $n=3$, the real unconditional constant of the basis of elementary
  matrices in $\Sch^\infty$ is $5/3$ and differs from its complex
  unconditional constant, $\sqrt3$. In fact, because the canonical
  bases of $\ell^2_\Col$ and $\ell^2_\Row$ are symmetric, the norm of
  a Schur multiplier by real signs turns out to equal the norm of one
  of the following three Schur multipliers:
  \begin{equation*}
    \begin{pmatrix}
      1&1&1\\1&1&1\\1&1&1
    \end{pmatrix},~
    \begin{pmatrix}
      1&1&1\\1&1&1\\1&1&-1
    \end{pmatrix}\text{or}~
    \begin{pmatrix}
      -1&1&1\\1&-1&1\\1&1&-1
    \end{pmatrix}.
  \end{equation*}
  The first one has norm 1: it defines the identity. The second one has
  the same norm as the Schur multiplier
  $\begin{pmatrix}1&1\\1&-1\end{pmatrix}$, which is $\sqrt2$, because
  the norm of that multiplier equals the norm of its tensor product by
  $\Id_{\ell_2^2}$, which is $
  \begin{pmatrix}
    1&1&1&1\\
    1&1&1&1\\
    1&1&-1&-1\\
    1&1&-1&-1\\
  \end{pmatrix}
  $.  
  By
  Prop.~\ref{transfer} for $\Gamma=\Z/3\Z$, the third one has the same
  norm as the Fourier multiplier $\varphi=(-1,1,1)$ on
  $\Ell^\infty(G)$, where $G=\{z\in\C:z^3=1\}$: as this multiplier acts by convolution with
  $f=-1+z+z^2$,
  its norm is $\|f\|_{\Ell^1(G)}$, that is
  \begin{equation*}
    \bigl( |-1+1+1| + |-1+\e^{2\iu\pi/3}+\e^{4\iu\pi/3}| +
    |-1+\e^{4\iu\pi/3}+\e^{2\iu\pi/3}| \bigr) /3= 5/3.
  \end{equation*}
  Complex interpolation yields that the real unconditional constant of
  the basis of elementary matrices is in fact strictly less than its
  complex counterpart in all $\Sch^p$ with $p\ne2$.
\end{exa}

\section{\texorpdfstring{\sloppy Varopoulos' characterisation of unconditional matrices in $\Sch^\infty$}{Varopoulos' characterisation of unconditional matrices in the space of bounded operators}}\label{V}\label{ss5}

\noindent Our results may be seen as the isometric counterpart to results by
Varopoulos \cite{va69} on tensor algebras over discrete spaces and
their generalisation to $\Sch^p$. He characterised unconditional basic
sequences of elementary matrices in $\Sch^\infty$ in his study of the
projective tensor product $\cnaught\projtens \cnaught$. We gather up his results
in the next theorem, as they are difficult to extract from the
literature.

\begin{thm}\label{varo}
  Let $I\subseteq\Row\times\Col$. The following are equivalent.
  \begin{enumerate}
  \item $I$ is an unconditional basic sequence in $\Sch^\infty$.
  \item $I$ is an \emph{interpolation set} for Schur multipliers on $\Sch^\infty$:
    every bounded sequence on $I$ is the restriction of a Schur multiplier on $\Sch^\infty$.
  \item $I$ is a \emph{$\Varopoulos$-Sidon set} as defined in
    \cite[Def.~4.1]{va69}: every null
    sequence on $I$ is the restriction of the sequence of coefficients of
    a tensor in $\cnaught(\Col)\projtens \cnaught(\Row)$.
  \item The coefficients of every tensor in
    $\ell^1_\Col\injtens\ell^1_\Row$ with support in
    $I$ form an absolutely convergent series.
  \item $(z_cz'_r)_ {(r,c)\in I}$ is a \emph{Sidon set} in the dual of
    $\T^{\Col}\times\T^{\Row}$, that is, an unconditional basic sequence in
    $\Cont(\T^{\Col}\times\T^{\Row})$.
  \item There is a constant $\lambda$ such that for all
    $\Row'\subseteq\Row$ and $\Col'\subseteq\Col$ with $n$ elements
    $\card[I\cap\Row'\times\Col']\le\lambda n$.
  \item $I$ is a finite union of forests.
  \item $I$ is a finite union of row sections and column sections.
  \item Every bounded sequence supported by $I$ is a Schur
    multiplier on $\Sch^\infty$.
  \end{enumerate}
  \end{thm}

\emph{Sketch of proof. } $(a)\Rightarrow(b)$. If $(a)$ holds, every sequence of signs
$\epsilon\in\{-1,1\}^I$ is a Schur multiplier on $\Sch^\infty_I$. By a
convexity argument, this implies that every bounded sequence is a
Schur multiplier on $\Sch^\infty_I$, which may be extended to a Schur
multiplier on $\Sch^\infty$ with the same norm by
\cite[Cor.~3.3]{pps89}.

$(b)\Rightarrow(c)$ holds by Grothendieck's inequality (see \cite[\S\,5]{pi01}) and an
approximation argument.

$(d)$ is but the formulation dual to $(c)$ (see \cite[\S\,6.2]{va67}.)

$(d)\Rightarrow(e)$. A computation yields
\begin{equation}\label{sidon}
  \biggl\|\sum_{(r,c)\in I}a_{rc}\e_c\otimes\e_r\biggr\|
  _{\ell^1_\Col\injtens\ell^1_\Row} =
  \sup_{|z_c|,|z'_r|=1}\biggl|\sum_{(r,c)\in I}a_{rc}z_cz'_r\biggr|.
\end{equation}

$(e)\Rightarrow(f)$ is \cite[Th.~4.2]{va69}. (The proof can be found in
\cite[\S\,6.3]{va67} and in \cite[\S\,5]{va68}.)

$(f)\Rightarrow(g)$, $(f)\Rightarrow(h)$ can be found in \cite[Th.~6.1]{va68}.

$(g)\Rightarrow(h)$. In fact, a forest is the union of a column section
$I'$ with a row section $I''$ (a \emph{bisection} in the
terminology of \cite[Def.~4.3]{va69}.) It suffices to prove this for a
tree. Let its vertices be indexed by words as described in the
Terminology. Then the set $I'$ of couples of the form $(w,w\concat c)$
with $w$ a word and $c$ a letter is a column section; the set $I''$ of
couples of the form $(w\concat r,w)$ with $w$ a word and $r$ a letter
is a row section.

$(h)\Rightarrow(i)$ is \cite[Th.~4.5]{va68}. Note that row sections
and column sections form 1-un\-con\-di\-tio\-nal basic sequences in
$\Sch^\infty$ and are 1-complemented in $\Sch^\infty$ by Prop.~\ref{complem}.

$(i)\Rightarrow(a)$ follows from the open mapping theorem.

\section{Closed walk relations}
\label{ss6}

\noindent We now introduce and study the combinatorial objects that we need in
order to analyse the expansion of the function defined by
\begin{equation}\label{Phidef}
  \Phi_I(\epsilon,a)= 
  \tr\Bigl|\sum_{q\in I}\epsilon_qa_q\e_q\Bigr|^p
\end{equation}
for $I\subseteq\Row\times\Col$, a positive even integer $p=2k$, signs $\epsilon_q\in\T$ and
coefficients $a_q\in\C$, of which only a finite number are nonzero. In fact,
\begin{equation}
  \label{compu}
\begin{split}
  \Phi_I(\epsilon,a)&=\tr\Bigl(\sum_{(r,c),(r',c')\in I}
  (\epsilon_{rc}a_{rc}\e_{rc})^*
  (\epsilon_{r'c'}a_{r'c'}\e_{r'c'})
  \Bigr)^k\\
  &=\tr~\sum_{\substack{(r_1,c_1),(r'_1,c'_1),\dots,\\
      (r_k,c_k),(r_k',c'_k)\in I}}
  ~\prod_{i=1}^k
  (\epsilon_{r_ic_i}^{-1}\overline{a_{r_ic_i}}\e_{c_ir_i})
  (\epsilon_{r'_ic'_i}a_{r'_ic'_i}\e_{r'_ic'_i})\\
  &=\sum_{\substack{(r_1,c_1),(r_1,c_2),\dots,\\
      (r_k,c_k),(r_k,c_{k+1})\in I}}
  ~\prod_{i=1}^k\epsilon_{r_ic_i}^{-1}\epsilon_{r_ic_{i+1}}
  \overline{a_{r_ic_i}}a_{r_ic_{i+1}}\quad(\text{where}~c_{k+1}=c_1.)
\end{split}
\end{equation}
The latter sum runs over all closed walks
$(c_1,r_1,c_2,\dots,c_k,r_k)$ of length $p$ in the graph $I$. With
multinomial notation, its terms have the form
$\epsilon^{\beta-\alpha}\overline{a}^\alpha a^\beta$ with
$|\alpha|=|\beta|=k$. The attempt to describe those couples
$(\alpha,\beta)$ that effectively arise in this expansion yields the
following definition.

\begin{dfn}\label{cwd}
  Let $p=2k\ge0$ be an even integer and $I\subseteq\Row\times\Col$. 

  \begin{enumerate}
  \item Let $\Alpha_k^I=\{\alpha\in\N^I:\sum_{q\in
      I}\alpha_q=k\}$ and set
    \begin{equation*}
      \textstyle
      \Beta_{k}^I=
      \bigl\{(\alpha,\beta)\in\Alpha_k^I\times\Alpha_k^I:
      \forall\, r\,\sum_c\alpha_{rc}=\sum_c\beta_{rc}~\text{and}~
      \forall\, c\,\sum_r\alpha_{rc}=\sum_r\beta_{rc}\bigr\}.
    \end{equation*}
  \item Two couples $(\alpha^1,\beta^1)\in\Beta_{k_1}^I$,
    $(\alpha^2,\beta^2)\in\Beta_{k_2}^I$ are \emph{disjoint} if
    $k_1,k_2\ge1$ and 
    \begin{equation}\label{alpha}
      \alpha^1_{rc}\ge1 \quad\Rightarrow\quad \forall\,(r',c)\in
      I\quad \alpha^2_{r'c}=0 \quad\text{and}\quad \forall\,(r,c')\in
      I\quad \alpha^2_{rc'}=0.
    \end{equation}
  \item The set $\WR_{k}^I$ of \emph{closed walk relations} of length $p$ in $I$ is
    the subset of those $(\alpha,\beta)\in\Beta_{k}^I$ that cannot be
    decomposed into the sum of two disjoint couples.
  \item Let $\Walk_k^I$ be the set of closed
    walks of length $p$ in the graph $I$. To every closed walk 
    $P=(c_1,r_1,c_2,r_2,\dots,c_k,r_k)$ of length $p$ we associate the
    couple $(\alpha,\beta)\in\Alpha_k^I\times\Alpha_k^I$ defined by 
    \begin{align*}
      \alpha_q&=\card\bigl[i\in\{1,\dots,k\}:(r_i,c_i)=q\bigr]\\
      \beta_q&=\card\bigl[i\in\{1,\dots,k\}:(r_i,c_{i+1})=q\bigr]
      \quad(\text{where}~c_{k+1}=c_1.)
    \end{align*}
    We shall write $P\sim (\alpha,\beta)$ and call $n_{\alpha\beta}$
    the number of elements of $\Walk_k^I$ mapped onto
    $(\alpha,\beta)$.
  \end{enumerate}
  
\end{dfn}

Note that the conditions in Eq.~\eqref{alpha} is in fact symmetric and
that it may be stated with $\beta^1$ and $\beta^2$ instead of $\alpha^1$
and $\alpha^2$.

\begin{exa}
  Let $\Row=\Col=\{0,1,2,3\}$ and $I=\Row\times\Col$. The couple
  $(\e_{00}+\allowbreak\e_{11}+\allowbreak\e_{22}+\allowbreak\e_{33},
  \allowbreak\e_{01}+\allowbreak\e_{10}+\allowbreak\e_{23}+\allowbreak\e_{32})$
  is an element of $\Beta_{4}^I\setminus\WR_{4}^I$: it is the sum of the
  two disjoint closed walk relations
  $(\e_{00}+\allowbreak\e_{11},\allowbreak\e_{01}+\allowbreak\e_{10})$
  and
  $\allowbreak(\e_{22}+\allowbreak\e_{33},\allowbreak\e_{23}+\allowbreak\e_{32})$.
\end{exa}

\begin{exa} Let $I=\Row\times\Col=\{0,1\}\times\{0,1\}$. Two closed
  walks are associated with the closed walk relation
  $(\e_{00}+\allowbreak \e_{11},\allowbreak \e_{01}+\allowbreak
  \e_{10})\in\WR_{2}^I$: the two cycles $(\text{col}\,0,\allowbreak
  \text{row}\,0,\allowbreak \text{col}\,1,\allowbreak \text{row}\,1)$
  and $(\text{col}\,1,\allowbreak \text{row}\,1,\allowbreak
  \text{col}\,0,\allowbreak \text{row}\,0)$.  Six closed walks are
  mapped onto the closed walk relation $(2\e_{00}+2\e_{01},\allowbreak
  2\e_{00}+2\e_{01})$: the $\dfrac{4!}{2!\,2!}$ concatenations of a
  permutation of $(\text{col}\,1,\allowbreak\text{row}\,0)$,
  $(\text{col}\,1,\allowbreak\text{row}\,0)$,
  $(\text{col}\,0,\allowbreak\text{row}\,0)$,
  $(\text{col}\,0,\allowbreak\text{row}\,0)$.
\end{exa}

The next proposition shows that, for our purpose, closed walk
relations describe entirely closed walks.

\begin{prp}
  \label{gon:bijection}
  Let $p=2k\ge0$ be an even integer and
  $I\subseteq\Row\times\Col$. The image of the mapping in
  Def.~\ref{cwd}$(d)$ is $\WR_{k}^I$:

  \begin{enumerate}
  \item if $P\in\Walk_k^I$ and $P\sim(\alpha,\beta)$, then
    $(\alpha,\beta)\in\WR_{k}^I$;
  \item if $(\alpha,\beta)\in\WR_{k}^I$, then there is a $P\in\Walk_k^I$ such that $P\sim(\alpha,\beta)$, so that
    $n_{\alpha\beta}\ge1$.
  \end{enumerate}

\end{prp}

\begin{proof} 
  $(a)$. Let $P=(c_1,r_1,c_2,r_2,\dots,c_k,r_k)$. In fact, 
  \begin{gather*}
    \sum\nolimits_c\alpha_{rc}=\card[i\in\{1,\dots,k\}:r_i=r]=\sum\nolimits_c\beta_{rc}\\
    \sum\nolimits_r\alpha_{rc}=\card[i\in\{1,\dots,k\}:c_i=c]=\sum\nolimits_r\beta_{rc}
  \end{gather*}
  and $(\alpha,\beta)\in\Beta_{k}^I$. If
  $(\alpha,\beta)=(\alpha^1,\beta^1)+(\alpha^2,\beta^2)$ with 
  $(\alpha^i,\beta^i)\in\Beta_{k_i}^I$ and $k_i\ge1$, there is an $i$ such
  that $\alpha^1_{r_ic_i}\ge1$ and $\alpha^2_{r_{i+1}c_{i+1}}\ge1$
  (where $(r_{i+1},c_{i+1})=(r_1,c_1)$ if $i=k$.) If
  $\beta^1_{r_ic_{i+1}}\ge1$, then $\sum_r\alpha^1_{rc_{i+1}}\ge1$, so
  that there is an $r$ such that
  $\alpha^1_{rc_{i+1}}\ge1$. Otherwise $\beta^2_{r_ic_{i+1}}\ge1$, so
  that $\sum_c\alpha^2_{r_ic}\ge1$ and there is a $c$ such that
  $\alpha^2_{r_ic}\ge1$. Therefore $(\alpha^1,\beta^1)$ and
  $(\alpha^2,\beta^2)$ are not disjoint and $(\alpha,\beta)\in\WR_{k}^I$.

  $(b)$. We have to find a closed walk of length $p$ that is mapped onto
  $(\alpha,\beta)$. If $k=0$, the
  empty closed walk suits.  Suppose that $k\ge1$; Consider a walk
  $(c_1,r_1,c_2,r_2,\dots,c_j,r_j,c_{j+1})$ in $I$ such that
  $\alpha^1_q=\card[i:(r_i,c_i)=q]\le\alpha_q$ and
  $\beta^1_q=\card[i:(r_i,c_{i+1})=q]\le\beta_q$ for every
  $q\in\Row\times\Col$, and furthermore $j$ is maximal.  We claim $(A)$
  that $c_{j+1}=c_1$ and $(B)$ that $j=k$. Let
  $(\alpha^2,\beta^2)=(\alpha,\beta)-(\alpha^1,\beta^1)$.

  $(A)$. If $c_{j+1}\ne c_1$, then
  \begin{gather*}
    \sum\nolimits_r\alpha^1_{rc_{j+1}}=\card[i\in\{1,\dots,j\}:c_i=c_{j+1}]\\
    \sum\nolimits_r\beta^1_{rc_{j+1}}=\card[i\in\{1,\dots,j+1\}:c_i=c_{j+1}] =
    1 + \sum\nolimits_r\alpha^1_{rc_{j+1}},
  \end{gather*}
  so that there must be $r_{j+1}$ with
  $\alpha^2_{r_{j+1}c_{j+1}}\ge1$. But then
  \[\sum\nolimits_c\beta^2_{r_{j+1}c}=\sum\nolimits_c\alpha^2_{r_{j+1}c}\ge1\] and there
  must be $c_{j+2}$ such that $\beta^2_{r_{j+1}c_{j+2}}\ge1$: $j$ is not
  maximal.

  $(B)$. Suppose that $j<k$. Then $(\alpha^1,\beta^1)\in\Beta_{j}^I$ and  
  $(\alpha^2,\beta^2)\in\Beta_{k-j}^I$. By hypothesis, they are not disjoint: there are $r,c,c'$ 
  such that $\alpha^1_{rc}\alpha^2_{rc'}\ge1$ or $r,r',c$ such
  that $\alpha^1_{rc}\alpha^2_{r'c}\ge1$. By interchanging $\Row$ and
  $\Col$ and by relabelling the vertices if necessary, we may suppose without loss of
  generality that for $r'_1=r_j$ there is $c'_1$ such that
  $\alpha^2_{r'_1c'_1}\ge1$. Then there is $c'_2$ such that
  $\beta^2_{r'_1c'_2}\ge1$. By the argument used in Claim $(A)$, there is a
  closed walk 
  $(c'_1,r'_1,c'_2,\dots,c'_{j'},r'_{j'})$ such that
  $\card[i:(r'_i,c'_i)=q]\le\alpha^2_q$ and
  $\card[i:(r'_i,c'_{i+1})=q]\le\beta^2_q$ (where $c'_{j'+1}=c'_1$.) 
  Then the closed walk 
  \begin{equation*}(c_1,r_1,c_2,r_2,\dots,c_j,r_j,c'_2,r'_2,
    \dots,c'_{j'},r'_{j'},c'_1,r'_1)\end{equation*}
  shows that $j$ is not maximal.  
\end{proof}

We are now in position to state the following theorem, a matrix counterpart to the computation
presented in \cite[Prop.~2.5$(ii)$]{ne98}.

\begin{thm}\label{sigma:ldc}
  Let $p=2k$ be a positive even integer and $I\subseteq\Row\times\Col$. 

  \begin{enumerate}
  \item The function $\Phi_I$ in Eq.~\eqref{Phidef} has the expansion
    \begin{equation}\label{gon:expand}
      \Phi_I(\epsilon,a) = 
      \sum_{(\alpha,\beta)\in\WR_{k}^I}
      n_{\alpha\beta}\epsilon^{\beta-\alpha}\overline{a}^\alpha
      a^\beta,
    \end{equation}
    where $n_{\alpha\beta}\ge1$ for every 
    $(\alpha,\beta)\in\WR_{k}^I$.
  \item If $\epsilon\in\T^I$ and $a\in(\Sch^p)^I$ is finitely
  supported, then the function
    \begin{equation}\label{gon:psi}
      \Psi_I(\epsilon,a) = \tr\Bigl|\sum_{q\in
          I}\epsilon_qa_q\e_q\Bigr|^p
    \end{equation}
    has the expansion
    \begin{equation}\label{gon:expand2}
    \sum_{(\alpha,\beta)\in\WR_{k}^I}
      \epsilon^{\beta-\alpha}\sum_{(c_1,r_1,\dots,c_k,r_k)\sim(\alpha,\beta)}
      \prod_{i=1}^ka_{r_ic_i}^*a_{r_ic_{i+1}}~(\text{with}~c_{k+1}=c_1.)
   \end{equation}
  \end{enumerate}

\end{thm}

\begin{proof} 
  This follows from Def.~\ref{cwd} and Prop.~\ref{gon:bijection}.
\end{proof}

Note that the edges of a closed walk $P\sim(\alpha,\beta)$
are precisely those $\{r,c\}$ such that $\alpha_{rc}+\beta_{rc}\ge1$.
$P$ is a cycle if and only if $P$ does not have length 0 or 2 and
$\sum_r\alpha_{rc}\le1$ for all $c$ and
$\sum_c\alpha_{rc}\le1$ for all $r$. We now show how to decompose
closed walks into cycles.

\begin{prp}\label{gon:dec}
  Let $P=(c_1,r_1,c_2,r_2,\dots,c_k,r_k)\sim(\alpha,\beta)$ be a closed walk.
  
  \begin{enumerate}
  \item If $r_i=r_j$ \klammer{vs.~$c_i=c_j$} for some
    $i\ne j$, then $P$ is the juxtaposition of two nonempty closed walks
    $P_1\sim(\alpha^1,\beta^1)$ and $P_2\sim(\alpha^2,\beta^2)$ such
    that $(\alpha,\beta)=(\alpha^1,\beta^1)+(\alpha^2,\beta^2)$ and
    $\sum_c\alpha^1_{r_ic}\,,\,\sum_c\alpha^2_{r_ic}\ge1$
    \klammer{vs.~$\sum_r\alpha^1_{rc_i}\,,\,\sum_r\alpha^2_{rc_i}\ge1$.}
  \item $P$ is the juxtaposition of nonempty closed walks
    $P_j\sim(\alpha^j,\beta^j)$ such that
    $\sum_r\alpha^j_{rc}\le1$ for all $c$,
    $\sum_c\alpha^j_{rc}\le1$ for all $r$ and 
    $(\alpha,\beta)=\sum(\alpha^j,\beta^j)$.
  \item There are cycles $P_j\sim(\alpha^j,\beta^j)$ and a $\gamma$ such that
    $(\alpha,\beta)=(\gamma,\gamma)+\sum(\alpha^j,\beta^j)$.
  \end{enumerate}
  
\end{prp}

\begin{proof}
  $(a)$. If $r_i=r_j$ for $i<j$, we may suppose that $j=k$: consider
  the closed walks 
  $P_1=(c_1,\allowbreak r_1,\dots,\allowbreak c_i,\allowbreak r_i)$ and
  $P_2=(c_{i+1},r_{i+1},\dots,c_k,r_k)$. If $c_i=c_j$ for $i<j$, we
  may suppose that $i=1$: consider then
  $P_1=(c_1,r_1,\dots,c_{j-1},r_{j-1})$ and
  $P_2=(c_j,r_j,\dots,c_k,r_k)$.
  
  $(b)$. Use $(a)$ in a maximality argument.
  
  $(c)$. Note that the closed walks $P_j$ in $(b)$ are either cycles
  or have length 2; in the latter case $P_j=q\sim(\e_q,\e_q)$ for some
  $q\in I$.
\end{proof}

\section{Schur multipliers on a cycle}\label{ss7}
\label{sec:cst}

\noindent We can realise a cycle of even length $2s$, $s\ge2$, in the following
convenient way. Let $\Gamma=\Z/s\Z$. Then the adjacency relation of
integers modulo $s$ turns $\Gamma$ into the cycle $(0,1,\dots,s-1)$ of
length $s$. We double this cycle into the bipartite cycle
$(\text{col}\,0,\allowbreak \text{row}\,0,\allowbreak
\text{col}\,1,\allowbreak \text{row}\,1,\dots,\allowbreak
\text{col}\,{s-1},\allowbreak \text{row}\,{s-1})$ on $\Gamma\amalg
\Gamma$, corresponding to the set of couples $I=\{(i,i),(i,i+1):i\in
\Gamma\}\subseteq \Gamma\times \Gamma$: $I$ is the pattern
\[
\bordermatrix{
  &\scriptstyle0&\scriptstyle1&\scriptstyle2&\scriptstyle\cdots&\scriptstyle s-2&\scriptstyle s-1\cr
  \hfil\scriptstyle0&*&*&0&\ddots&0&0\cr
  \hfil\scriptstyle1&0&*&*&\ddots&0&0\cr
  \hfil\scriptstyle2&0&0&*&\ddots&0&0\cr
  \hfil\scriptstyle\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\cr
  \hfil\scriptstyle s-2&0&0&0&\ddots&*&*\cr
  \hfil\scriptstyle s-1&*&0&0&\ddots&0&*\cr}\text.
\]
$\Gamma$ is the group dual to $G=\hat \Gamma=\{z\in\C:z^s=1\}$.  We
shall consider the space $\Ell^p_\Lambda(G)$ spanned by
$\Lambda=\{1,z\}$ in $\Ell^p(G)$, where $z$ is the identical function
on $G$: its norm is given by
$\|a+bz\|_{\Ell^p(G)}=\big(s^{-1}\sum_{z^s=1}|a+bz|^p\bigr)^{1/p}$.

\begin{prp}
  \label{cyno}
  Let $0<p\le\infty$, $s\ge2$ and $I=\{(i,i),(i,i+1):i\in\Z/s\Z\}$.
  Let $\epsilon\in\T^I$ be a Schur multiplier by signs on
  $\Sch^p_I$.
  
  \begin{enumerate}
  \item The Schur multiplier $\epsilon$ has the same norm as the Schur multiplier
    $\hat\epsilon$ given by $\hat\epsilon_q=1$ for
    $q\ne({s-1},0)$ and $\hat\epsilon_{s-1,0} =
    \overline{\epsilon_{00}} \epsilon_{01} \dots
    \overline{\epsilon_{{s-1},{s-1}}}
    \epsilon_{{s-1},0}$.
  \item The Schur multiplier $\epsilon$ has the same norm as
    $\check\epsilon$ given by $\check\epsilon_{ii} = 1$ and
    $\check\epsilon_{i,i+1}=\vartheta$ with $\vartheta$ any
    $s$th root of $\hat\epsilon_{{s-1},0}$ or its complex
    conjugate: without loss of generality, $\vartheta=\e^{\iu\alpha}$
    with $\alpha\in[0,\pi/s]$.
  \item The norm of $\epsilon$ on $\Sch^p_I$ is bounded below by the
    norm of the relative Fourier multiplier $\mu:a+bz\mapsto
    a+\vartheta bz$ on $\Ell^p_\Lambda(G)$; their complete norms are equal.
  \item The norm of $\epsilon$ on $\Sch^1_I$ and on
    $\Sch^\infty_I$ is equal to the norm of $\mu$ on
    $\Ell^1_\Lambda(G)$ and on $\Ell^\infty_\Lambda(G)$: this norm is
    \begin{equation*}
      \displaystyle\frac{\cos(\alpha/2-\pi/2s)}{\cos\pi/2s}
      =\frac{\max_{z^s=-1}|\vartheta+z|}{|1+\e^{\iu\pi/s}|}\text.
    \end{equation*}
  \item The Schur multiplication operator $\Schur_\epsilon$ is an
    isometry on $\Sch^p_I$ if and only if
    $p/2\in\{1,2,\dots,s-1\}$ or $\overline{\epsilon_{00}}
    \epsilon_{01} \dots \overline{\epsilon_{{s-1},{s-1}}}
    \epsilon_{{s-1},0}=1$.
  \end{enumerate}
  
\end{prp}

\begin{proof}
  $(a)$ and $(b)$ follow from the matrix unconditionality of
  Schatten-von-Neu\-mann norms (see Eq.~\eqref{mu}) and from the fact
  that the Schur multipliers $\epsilon$ and
  $\bar\epsilon=(\overline{\epsilon_q})_{q\in I}$ have the same norm
  on $\Sch^p_I$.
  
  $(c)$ follows from Prop.~\ref{transfer}.

  $(d)$.  Let us compute
  $f(\beta)={\|1+\e^{\iu\beta}z\|}_{\Ell^1(G)}$. As
  $f(\beta)=f(\beta+2\pi/s)=f(-\beta)$, we may suppose without loss of
  generality that $\beta\in[0,\pi/s]$. Then
  $\lvert\beta/2+k\pi/s\rvert\le\pi/2$ if $-\lfloor s/2\rfloor\le
  k\le\lceil s/2\rceil-1$, so that
  \begin{align*}
    f(\beta)
    &=\frac1s\sum_{k=-\lfloor s/2\rfloor}^{\lceil s/2\rceil-1}
    \bigl\lvert1+\e^{\iu\beta}\e^{2\iu k\pi/s}\bigr\rvert\\
    &=\frac2s\sum_{k=-\lfloor s/2\rfloor}^{\lceil s/2\rceil-1}
    \cos(\beta/2+k\pi/s)\\
    &=\frac2s\Re\biggl(\e^{\iu\beta/2}\frac{\e^{\iu\lceil s/2\rceil\pi/s}-\e^{-\iu\lfloor s/2\rfloor/s}}{\e^{\iu\pi/s}-1}\biggr)\\
    &=\frac2{s\sin(\pi/2s)}\cdot
    \begin{cases}
      \cos(\beta/2-\pi/2s)&\text{if $s$ is even}\\
      \cos(\beta/2)&\text{if $s$ is odd.}
    \end{cases}
  \end{align*}
  This shows in both cases that the norm of $\mu$ on $\Ell^1_\Lambda(G)$ is bounded
  below by $\cos(\alpha/2-\pi/2s)\slash\cos(\pi/2s)$. 
  The complete norm of $\mu$ on $\Ell^\infty_\Lambda(G)$ is equal to
  its norm and thus to the maximum of
  \begin{math}
    g(w)
    ={\|w+\vartheta z\|}_{\Ell^\infty(G)}\slash{\|w+z\|}_{\Ell^\infty(G)}
  \end{math}
  for $w\in\C$. Let $w=r\e^{\iu\beta}$ with $r\ge0$ and $\beta\in\R$.
  Note that
  \begin{equation*}
    {\|w+z\|}_{\Ell^\infty(G)}=\bigl\lvert r+\e^{\iu\mathrm{d}(\beta,(2\pi/s)\Z)}\bigr\rvert
  \end{equation*}
  is a decreasing function of $\mathrm{d}(\beta,(2\pi/s)\Z)$ and that
  \begin{equation*}
    \mathrm{d}(\alpha-\beta,(2\pi/s)\Z)<\mathrm{d}(\beta,(2\pi/s)\Z)
    \ \Leftrightarrow\ 
    \beta\in\mathopen]\alpha/2,\pi/s+\alpha/2\mathclose[\bmod{2\pi/s}\text.
  \end{equation*}
  As
  $g(w)=g(wz)$ if $z^s=1$, we may suppose without loss of
  generality that $\beta\in\mathopen]\alpha/2,\pi/s+\alpha/2\mathclose[$. Therefore
  \begin{equation*}
    g(w)=
    \begin{cases}
      \Bigl|\dfrac{w+\e^{\iu\alpha}}{w+1}\Bigr|&\text{if $\beta\in\mathopen]\alpha/2,\pi/s]$}\\
      \Bigl|\dfrac{w+\e^{\iu\alpha}}{w+\e^{2\iu\pi/s}}\Bigr|&\text{if
        $\beta\in[\pi/s,\pi/s+\alpha/2\mathclose[$.}
    \end{cases}
  \end{equation*}
  As $g$ tends to 1 at infinity and $g(w)=1$ if
  $\beta\in\{\alpha/2,\pi/s+\alpha/2\}$, the maximum principle shows
  that $g$ attains its maximum with $\beta=\pi/s$. Finally,
  \begin{align*}
    {g(r\e^{\iu\pi/s})}^2
    &=\frac{1+2r\cos(\pi/s-\alpha)+r^2}{1+2r\cos(\pi/s)+r^2}\\
    &=1+\frac{\cos(\pi/s-\alpha)-\cos\pi/s}{\cos(\pi/s)+(r+1/r)/2}
    \le{g(\e^{\iu\pi/s})}^2
    ={\biggl(\frac{\cos(\pi/2s-\alpha/2)}{\cos\pi/2s}\biggr)}^2\text.
  \end{align*}
    $(e)$. If $p$ is not an even integer and $\vartheta^s\ne1$, then
  $\mu$ is not an isometry on $\Ell^p_\Lambda(G)$: otherwise the
  functions $z$ and $\vartheta z$ would have the same distribution by
  the Plotkin-Rudin Equimeasurability Theorem (see \cite[Th.~2]{kk01}). If
  $p\in\{2,4,\dots,2s-2\}$, then $I$ contains no cycle of length
  $4,6,\dots,p$, so that by Prop.~\ref{gon:dec}$(c)$ every closed walk
  $P\sim(\alpha,\beta)$ satisfies $\alpha=\beta$. The function
  $\Phi_I(\epsilon,a)$ in Eq.~\eqref{Phidef} is therefore constant in
  $\epsilon$ by Th.~\ref{sigma:ldc}$(a)$. If $p\in\{2s,2s+2,\dots\}$,
  the closed walk relation
  \begin{equation*}
    (\alpha,\beta) = \Bigl(\sum_{i\in \Gamma}\e_{ii}, \sum_{i\in
      \Gamma}\e_{i,i+1}\Bigr)+(p/2-s)(\e_{00},\e_{00})
  \end{equation*}
  satisfies $n_{\alpha\beta}\ge1$ by Prop.~\ref{gon:bijection}. Then
  the coefficient of $\Phi_I(\epsilon,a)$ in ${\bar a}^\alpha a^\beta$
  equals
  \begin{equation*}
    n_{\alpha\beta}\overline{\epsilon_{00}} \epsilon_{01} \dots
  \overline{\epsilon_{{s-1},{s-1}}}\epsilon_{{s-1},0}
\end{equation*}
  and must equal
  the same quantity with $\epsilon$ replaced by 1 if $\epsilon$
  defines an isometry on $\Sch^p_I$.
\end{proof}

\begin{rem} 
  See \cite[p.~245]{li96} for a similar application of the
  Plotkin-Rudin Equimeasurability Theorem in $(e)$.
\end{rem}

The real unconditional constant of $I$ is therefore the norm of
$\check\epsilon$ with $\alpha=\pi/s$, and the complex
unconditional constant is the maximum of the norm of $\check\epsilon$ for
$\alpha\in[0,\pi/s]$.
This yields

\begin{cor}\label{cyc}
  Let $0<p\le\infty$ and $s\ge2$. Let $I$ be the cycle of
  length $2s$.

  \begin{enumerate}
  \item $I$ is a real 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$
    if and only if $p\in\{2,\allowbreak 4,\dots,\allowbreak 2s-2\}$. 
  \item The real and complex unconditional constants of $I$
    in the spaces $\Sch^1$ and $\Sch^\infty$ equal $\sec\pi/2s$.
  \end{enumerate}

\end{cor}

\section{\texorpdfstring{1-unconditional matrices in $\Sch^p$,  $p$ not an even integer}{1-unconditional matrices in p-Schatten classes,  p not an even integer}}\label{ss8}

\noindent We now state the announced isometric counterpart to Varopoulos'
characterisation of unconditional matrices in $\Sch^\infty$ (Section \ref{V})
and its generalisation to $\Sch^p$ for $p$ not an even integer.

\begin{thm}
  \label{p-np}
  Let $I\subseteq\Row\times\Col$ be nonempty and
  $p\in(0,\infty]\setminus2\N$.  The following are
  equivalent.
  
  \begin{enumerate}
  \item $I$ is a complex completely 1-un\-con\-di\-tio\-nal basic
    sequence in $\Sch^p$.
  \item $I$ is a complex 1-un\-con\-di\-tio\-nal basic sequence in
    $\Sch^p$.
  \item $I$ is a real 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$.
  \item $I$ is a forest.
  \item For each $\epsilon\in\T^I$ there are
    $\zeta\in\T^\Col$  and $\eta\in\T^\Row$ such that
    $\epsilon_{rc}=\zeta(c)\eta(r)$ for all $(r,c)\in I$.
  \item For each $\epsilon\in\{-1,1\}^I$ there are
    $\zeta\in\{-1,1\}^\Col$  and $\eta\in\{-1,1\}^\Row$ such that
    $\epsilon_{rc}=\zeta(c)\eta(r)$ for all $(r,c)\in I$.
  \item $I$ is a \emph{set of $\Varopoulos$-interpolation of constant 1}: for all
    $\varphi\in\ell^\infty_I$ 
    \begin{equation}
      \label{interpol}
      \inf\Bigl\{\biggl\|
      \sum_{(r,c)\in\Row\times\Col}
      \tilde{\varphi}_{rc}\e_c\otimes\e_r
      \biggr\|_{\ell^\infty_\Col\projtens \ell^\infty_\Row}:
      \tilde{\varphi}|_I=\varphi\Bigr\}=\sup_{q\in I}|\varphi_q|.
    \end{equation}
  \item $I$ is a \emph{$\Varopoulos$-Sidon set of constant 1}: for all
    $\varphi\in \cnaught(I)$ 
    \begin{equation}
      \label{sid}
      \inf\Bigl\{\biggl\|
      \sum_{(r,c)\in\Row\times\Col}
      \tilde{\varphi}_{rc}\e_c\otimes\e_r
      \biggr\|_{\cnaught(\Col)\projtens \cnaught(\Row)}:
      \tilde{\varphi}|_I=\varphi\Bigr\}=\sup_{q\in I}|\varphi_q|.
    \end{equation}
  \item For every tensor $u=\sum_{(r,c)\in
      I}a_{rc}\e_c\otimes\e_r$ in
    $\ell^1_\Col\injtens\ell^1_\Row$ with support in $I$
    we have
    $\|u\|_{\ell^1_\Col\injtens\ell^1_\Row}=\sum_{(r,c)\in
      I}|a_{rc}|$.
  \item $(z_cz'_r)_{(r,c)\in I}$ is a \emph{Sidon set of constant 1} in the
    dual of $\T^{\Col}\times\T^{\Row}$, that is, a 1-un\-con\-di\-tio\-nal basic
    sequence in $\Cont(\T^{\Col}\times\T^{\Row})$: if $(a_{rc})$ is
    finitely supported,
    \[\sup_{(z,z')\in\T^\Col\times\T^\Row}
    \Bigl|\sum_{(r,c)\in I}a_{rc}z_cz'_r\Bigr| = \sum_{(r,c)\in
    I}|a_{rc}|\ .\]
  \item For all $\Row'\subseteq\Row$ and $\Col'\subseteq\Col$ with
    $k\ge1$ elements $\card[I\cap\Row'\times\Col']\le2k-1$.
  \item $I$ is an \emph{isometric interpolation set} for Schur multipliers on
    $\Sch^\infty$: every $\varphi\in\ell^\infty_I$ is the restriction of a
    Schur multiplier on $\Sch^\infty$ with norm
    $\|\Schur_\varphi\|=\|\varphi\|_{\ell^\infty_I}$.
  \end{enumerate}

\end{thm}

\begin{proof} 
  $(a)\Rightarrow(b)\Rightarrow(c)$ is trivial.

  $(c)\Rightarrow(d)$. Suppose that $I$ contains a cycle
  $(c_0,r_0,\dots,c_{s-1},r_{s-1})$ with
  $s\ge2$. Cor.~\ref{cyc}$(a)$ shows
  that $I$ is not a real 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$.


  $(d)\Leftrightarrow(k)$. A tree on $2k$ vertices has exactly $2k-1$ edges, so
  that a forest $I$ satisfies $(k)$. Conversely, a cycle of length $2s$
  is a graph with $s$ row vertices, $s$ column vertices and $2s$
  edges.
  
  $(d)\Rightarrow(e)$. Suppose first that $I$ is a tree and index the vertices of its
  edges by words $w\in W$ as described in the Terminology. Let us
  define $\eta$ and $\zeta$ inductively. If $r$ is the root
  of the tree, indexed by $\emptyset$, let $\eta(r)=1$. Suppose that $\eta$ and
  $\zeta$ have been defined for all vertices indexed by words of
  length at most $2n$. If $c$ is indexed by a word $w$ of length
  $2n+1$, let $r$ be the vertex indexed by the word of length $2n$
  with which $w$ begins and let $\zeta(c)=\epsilon(r,c)\slash\eta(r)$. If $r$ is
  indexed by a word $w$ of length $2n+2$, let $c$ be the vertex
  indexed by the word of length $2n+1$ with which $w$ begins and let
  $\eta(r)=\epsilon(r,c)\slash\zeta(c)$.  If $I$ is a union of pairwise
  disjoint trees, we may define $\eta$ and $\zeta$ on each tree
  separately. We may finally extend $\eta$ to $\Row$ and $\zeta$ to
  $\Col$ in an arbitrary manner.
    
  $(d)\Rightarrow(f)$ may be proved as $(d)\Rightarrow(e)$.
  
  $(f)\Rightarrow(c)$. If $(f)$ holds, then every Schur multiplier by
  signs $\epsilon\in\{-1,1\}^I$ is elementary in the sense that
  $\epsilon=\zeta\otimes\eta$. The complete norm of $\Schur_\epsilon$ on
  any $\Sch^p_I$ is therefore
  $\|\zeta\|_{\ell^\infty_\Col}\|\eta\|_{\ell^\infty_\Row}=1$.
  
  $(e)\Rightarrow(g)$. If $(e)$ holds, every
  $\varphi\in\T^I\subseteq\ell^\infty_I$ may be extended to an
  elementary tensor $\zeta\otimes\eta$ of norm 1. $(g)$ follows
  because every element of $\ell^\infty_I$ with norm 1 is the half sum
  of two elements of $\T^I$: note that $\e^{\iu t}\cos u =
  \bigl(\e^{\iu(t+u)}+\e^{\iu(t-u)}\bigr)/2$.

  $(g)\Rightarrow(h)$. It suffices to check Equality~\eqref{sid} for $\varphi$ with support
  contained in a finite rectangle set $\Row'\times\Col'$. As $\ell^\infty_{\Col'}\projtens \ell^\infty_{\Row'}$
  is a subspace of $\ell^\infty_\Col\projtens \ell^\infty_\Row$,
  Eq.~\eqref{interpol} yields Eq.~\eqref{sid}.

  $(h)\Leftrightarrow(i)$ because they are dual statements.

  $(i)\Leftrightarrow(j)$. Use Equality \eqref{sidon}.

  $(h)\Rightarrow(l)$ may be deduced by the argument of
  Prop.~\ref{complem}$(a)\Rightarrow(b)$.

  $(l)\Rightarrow(a)$. Taking sign sequences $\varphi\in\T^I$ in $(l)$ shows that
  all relative Schur multipliers by signs on
  $\Sch^\infty_I$ define isometries. Apply Prop.~\ref{pi}. \end{proof}

\begin{rem}
  The equivalence of $(e)$ with $(j)$ may also be shown as a consequence of the
  characterisation of Sidon sets of constant 1 in \cite{chm81}.
\end{rem}

Let us now answer Question~\ref{q3}.

\begin{cor}
  Let $I\subseteq\Row\times\Col$. The following are equivalent.

  \begin{enumerate}
  \item For all $\varphi\in \cnaught(I)$ one has $\bigl\|\sum_{(r,c)\in I}
    \varphi_{rc} \e_c \otimes \e_r
    \bigr\|_{\cnaught(\Col)\projtens \cnaught(\Row)}=\sup_{q\in I}|\varphi_q|$.
  \item There are pairwise disjoint sets $\Row_j\subseteq\Row$ and pairwise disjoint sets
    $\Col_j\subseteq\Col$ such that $\Row_j$ or $\Col_j$ is a singleton for each $j$
    and $I=\bigcup\Row_j\times\Col_j$: $I$ is the union of the column
    section $\bigcup_{\card\Row_j=1}\Row_j\times\Col_j$ with the
    disjoint row section $\bigcup_{\card\Row_j>1}\Row_j\times\Col_j$.
  \item $I$ is a union of pairwise disjoint star graphs: every path in $I$ has
    length at most 2.
  \end{enumerate} 
  
\end{cor}

\begin{proof} 
  $(a)\Rightarrow(b)$ follows from
  Prop.~\ref{complem}$(a)\Rightarrow(d)$ and
  Th.~\ref{p-np}$(g)\Rightarrow(d)$.

  $(b)\Leftrightarrow(c)$. $(b)$ holds if and only if $(r,c),(r',c),(r,c')\in
  I\Rightarrow(r=r'~\text{or}~c=c')$ and therefore if and only if $(c)$ holds.

  $(b)\Rightarrow(a)$. Suppose $(b)$ and let $\varphi\in \cnaught(I)$.
  Let $\alpha_j=\sup_{(r,c)\in\Row_j\times\Col_j}|\varphi_{rc}|^{1/2}$. If
  $\alpha_j=0$, let us define $\varrho^j=0$ and $\gamma^j=0$. Otherwise, if
  $\Row_j$ is a singleton $\{r\}$, let us define $\varrho^j=\alpha_j\e_r$ and
  $\gamma^j$ by $\gamma^j_c=\varphi_{rc}/\alpha_j$ if $c\in\Col_j$ and
  $\gamma^j_c=0$ otherwise.  Otherwise, $\Col_j$ is a singleton $\{c\}$ and we
  define $\gamma^j=\alpha_j\e_c$ and $\varrho^j$ by
  $\varrho^j_r=\varphi_{rc}/\alpha_j$ if $r\in\Row_j$ and $\varrho^j_r=0$ otherwise.
  Note that the $\gamma^j$ have pairwise disjoint support and are null sequences,
  as well as the $\varrho^j$. Then
  \begin{equation*}
    \sum_{(r,c)\in I}\varphi_{rc}\e_c\otimes\e_r = 
    \sum_j\gamma^j\otimes\varrho^j = 
    \average_{\epsilon_j=\pm1}
    \biggl(\sum_j\epsilon_j\gamma^j\biggr)
    \otimes\biggl(\sum_j\epsilon_j\varrho^j\biggr)
  \end{equation*}
  is an average of elementary tensors in $\cnaught(\Col)\projtens \cnaught(\Row)$
  of norm $\sup_{q\in I}|\varphi_q|$, so that this average 
  is also 
  bounded by this norm, which obviously is a lower bound. 
\end{proof}

\section{\texorpdfstring{1-unconditional
    matrices in $\Sch^p$, $p$ an even integer}{1-unconditional
    matrices in p-Schatten classes, p an even integer}}
\label{ss9}

\noindent Let us now prove Theorem~\ref{th5} as a consequence of 
Theorem \ref{sigma:ldc} together with Proposition \ref{gon:dec}$(c)$.

\begin{thm}\label{sigma:THM}
  Let $I\subseteq\Row\times\Col$ and $p=2k$ a positive even integer. The following
  assertions are equivalent.
  
  \begin{enumerate}
  \item $I$ is a complex completely 1-un\-con\-di\-tio\-nal basic sequence in
    $\Sch^p$. 
  \item $I$ is a complex 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$.
  \item For every finite subset $F\subseteq I$ there is an operator $x\in
    \Sch^p$, whose support $S$ contains $F$, such that
    $\bigl\|\sum\epsilon_qx_q\e_q\bigr\|_p$ does not depend on the complex choice
    of signs $\epsilon\in\T^S$.
  \item $I$ is a real
    1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$.
  \item For every finite subset $F\subseteq I$ there is an operator $x\in
    \Sch^p$ with real matrix coefficients, whose support $S$ contains
    $F$, such that $\bigl\|\sum\epsilon_qx_q\e_q\bigr\|_p$ does not depend on the
    real choice of signs $\epsilon\in\{-1,1\}^S$.
  \item Every closed walk $P\sim(\alpha,\beta)$ of length $2s\le
    2k$ in $I$ satisfies $\alpha=\beta$.
  \item $I$ does not contain any cycle of length $2s\le 2k$ as a subgraph.
  \item For each $v,w\in V$ there is at most one path in $I$ of
    length $l\le k$ that joins $v$ to $w$.
  \end{enumerate}
  
\end{thm}

\begin{proof} 
  $(a)\Rightarrow(b)\Rightarrow(c)$, $(b)\Rightarrow(d)\Rightarrow(e)$ are trivial.
  
  $(c)\Rightarrow(g)$. Suppose that $I$ contains a cycle
  $P\sim(\gamma,\delta)$ of length $2s\le 2k$: the corresponding set
  of couples is $F=\{q:\gamma_q+\delta_q=1\}$. Let $x$ be as in $(c)$
  and let $(\alpha,\beta)=(\gamma,\delta)+(k-s)(\e_q,\e_q)$ for some
  arbitrary $q\in F$.  Then $(\alpha,\beta)\in\WR_{k}^S$. Consider
  $f(\epsilon)=\bigl\|\sum\epsilon_qx_q\e_q\bigr\|_p^p$ as a function
  on the group $\T^S$. Then the Fourier coefficient
  $\widehat{f}(\epsilon^{\beta-\alpha})$ of $f$ at the Steinhaus
  character $\epsilon^{\beta-\alpha}$ is, by Th.~\ref{sigma:ldc}$(a)$,
  \begin{multline*}
    \sum\bigl\{n_{\varepsilon\zeta}\overline{x}^\varepsilon x^\zeta:
    (\varepsilon,\zeta)\in\WR_{k}^S~\text{and}~
    \zeta-\varepsilon=\beta-\alpha\bigr\}\\
    =\overline{x}^\gamma x^\delta
    \sum\bigl\{n_{\varepsilon\zeta}\overline{x}^{\varepsilon-\gamma}
    x^{\zeta-\delta}:
    (\varepsilon,\zeta)\in\WR_{k}^S~\text{and}~
    \zeta-\delta=\varepsilon-\gamma\bigr\}.
  \end{multline*}
  (Note that $\beta-\alpha=\delta-\gamma$.) As this last
  sum has only positive terms and contains at least the term corresponding to
  $(\alpha,\beta)$, $f$ cannot be constant.

  $(e)\Rightarrow(g)$. Let $P\sim(\gamma,\delta)$, $F=\{q:\gamma_q+\delta_q=1\}$ and
  $(\alpha,\beta)$ be as in the proof of the implication $(c)\Rightarrow(h)$. Let $x$ be
  as in $(e)$.  Consider $f(\epsilon)=\bigl\|\sum\epsilon_qx_q\e_q\bigr\|_p^p$ as a
  function on the group $\{-1,1\}^S$. Then the Fourier coefficient
  $\widehat{f}(\epsilon^{\beta-\alpha})$ of $f$ at the
  Walsh character $\epsilon^{\beta-\alpha}$ is, by Th.~\ref{sigma:ldc}$(a)$,
  \begin{multline*}
    \sum\bigl\{n_{\varepsilon\zeta}x^{\varepsilon+\zeta}:
    (\varepsilon,\zeta)\in\WR_{k}^S~\text{and}~
    \zeta-\varepsilon\equiv\beta-\alpha\pmod{2}\bigr\}\\
    =x^{\gamma+\delta}
    \sum\bigl\{n_{\varepsilon\zeta}x^{\varepsilon+\zeta-\gamma-\delta}:
    (\varepsilon,\zeta)\in\WR_{k}^S~\text{and}~
    \zeta-\varepsilon\equiv\delta-\gamma\pmod{2}\bigr\}.
  \end{multline*}
  As this last sum has only positive terms and contains at least the term
  corresponding to $(\alpha,\beta)$, $f$ cannot be constant.

  $(f)\Leftrightarrow(g)$. Apply Prop.~\ref{gon:dec}$(c)$.


  $(g)\Leftrightarrow(h)$. If $I$ contains a cycle $(v_0,\dots,v_{2s-1})$, then $I$
  contains two distinct paths $(v_0,\dots,v_{s})$,
  $(v_0,v_{2s-1},\dots,v_{s})$ of length $s$ from $v_0$ to $v_{s}$. If
  $I$ contains two distinct paths $(v_0,\dots,v_l)$,
  $(v'_0,\dots,v'_{l'})$ with $v_0=v'_0$, $v_{l}=v'_{l'}$ and $l,l'\le
  k$, let $a$ be minimal such that $v_a\ne v'_a$, let $b\ge a$ be minimal such
  that $v_b\in\{v'_a,\dots,v'_{l'}\}$ and let $d\ge a$ be minimal such that
  $v'_d=v_b$. Then $(v_{a-1},\dots,v_b,v'_{d-1},\dots,v'_a)$ is a cycle
  in $I$
  of length $2s\le2k$.

  $(f)\Rightarrow(a)$ holds by Theorem~\ref{sigma:ldc}$(b)$: If each
  $(\alpha,\beta)\in\WR_{k}^I$ satisfies $\alpha=\beta$, then
  Eq.~\eqref{gon:expand2} shows that 
  $\Psi_I(\epsilon,z)$ as defined in
  Eq.~\eqref{gon:psi} is constant in $\epsilon$.
\end{proof}

\begin{rem}
  The equivalence $(b)\Leftrightarrow(g)$ is a noncommutative
  analogue to \cite[Prop.~2.5$(ii)$]{ne98}.
\end{rem}

\begin{rem}
  In \cite[Th.~2.7]{ne99}, the condition of Th.~\ref{sigma:THM}$(f)$
  is visualised in another way: a closed walk
  $P=(c_1,r_1,\dots,c_s,r_s)\sim(\alpha,\beta)$ in $\N\times\N$ is
  considered as the polygonal closed curve $\gamma$ in $\C$ with sides
  parallel to the coordinate axes whose successive vertices are
  $r_1+\iu c_1$, $r_1+\iu c_2$, $r_2+\iu c_2$, \dots, $r_{s-1}+\iu
  c_s$, $r_s+\iu c_s$, $r_s+\iu c_1$ and again $r_1+\iu c_1$. Then
  $\alpha=\beta$ if and only if the index with respect to $\gamma$ of
  every point not on $\gamma$ is zero, if and only if $\gamma$ can be
  shrunk to a point inside of the set of its points. 
\end{rem}

\begin{rem} 
  One cannot drop the assumption that $x$ has real matrix
  coefficients in Th.~\ref{sigma:THM}$(e)$. Consider a $2\times2$
  matrix $x$. Then $\tr x^*x=\sum|x_q|^2$ and $\det
  x^*x=|x_{00}x_{11}-x_{01}x_{10}|^2$.  This shows that if
  $\Re(\overline{x_{00}x_{11}}x_{01}x_{10})=0$, e.g.\ $x=
  \begin{pmatrix}
    1&1\\1&\iu
  \end{pmatrix}
  $, then the singular values of $x$ do not depend on the real sign of
  the matrix coefficients of $x$, whereas
  $(\text{col}\,0,\text{row}\,0,\text{col}\,1,\text{row}\,1)$ is a
  cycle of length 4.
\end{rem}

\begin{rem}
  Theorem~\ref{sigma:THM}$(h)\Rightarrow(a)$ is the isometric counterpart to
  \cite[Th.\ 3.1]{hno01}, which shows in particular that $I$ is an
  unconditional basic sequence in $\Sch^{2k}$ if the number of
  walks in $I$ between two given vertices of length $k$ and with no edge repeated
  has a uniform bound. The following combinatorial problem arises
  naturally: if $I$ satisfies this latter condition, is it
  so that $I$ is the union of a finite number of sets $I_j$ such that there
  is at most one path of length at most $k$ in $I_j$ between two given
  vertices? In the simplest case, $k=2$, William Banks, Ilijas Farah,
  Asma Harcharras and Dominique
  Lecomte \cite{fl03} have deduced from \cite{ro85} that it is not so.
\end{rem}

\section{\texorpdfstring{Metric unconditional approximation pro\-per\-ty for $\Sch^p_I$}{Metric unconditional approximation property for entry subspaces of a Schatten class}}\label{ss10}

\noindent Let $\Row,\Col$ be two copies of $\N$. It is well known that, apart
from $\Sch^2$, no $\Sch^p$
has an unconditional basis or just a local unconditional structure
(see \cite[\S\,4]{px03}.) $\Sch^1$ and $\Sch^\infty$ cannot even be
embedded in a space with unconditional basis. If $1<p<\infty$, then
$\Sch^p$ has the unconditional finite dimensional decomposition
\begin{equation*}
  \bigoplus_{n\in\N}\Sch^p_{\{(r,c):r\le n,c=n\}}\oplus
  \Sch^p_{\{(r,c):r=n+1,c\le n\}}
\end{equation*}
because the triangular projection associated to the idempotent Schur multiplier
$(\chi_{r\le c})$ is bounded on $\Sch^p$.

\begin{dfn}\label{muap:def}
  Let $X$ be a separable Banach space and $\Sign=\T$
  (vs.~$\Sign=\{-1,1\}$.)

  \begin{itemize}
  \item A sequence $(T_k)$ of operators on $X$ is an
    \emph{approximating sequence} if each $T_k$ has finite rank and
    $\|T_kx-x\|\to0$ for every $x\in X$. 
    An approximating sequence of commuting projections
    is a \emph{finite-dimensional decomposition}.
  \item (\cite{pw71}.) The \emph{difference sequence}
    $(\Delta T_k)$ of $(T_k)$ is given by $\Delta T_1=T_1$ and $\Delta
    T_k=T_k-T_{k-1}$ for $k\ge2$. $X$ has the \emph{unconditional
      approximation property} $(uap)$ if there is an approximating
    sequence $(T_k)$ such that for some constant~$D$
    \begin{equation*}
      \biggl\|\sum_{k=1}^n\epsilon_k\Delta T_k\biggr\|
      \le D\quad\text{for all $n$ and $\epsilon_k\in\Sign$}.
    \end{equation*}
    The \emph{complex \klammer{vs.~real} unconditional constant} of
    $(T_k)$ is the least such constant~$D$.
  \item (\cite[\S\,3]{ck91}, \cite[\S\,8]{gks93}.) $X$ has the
    \emph{complex \klammer{vs.~real} metric unconditional approximation
      property} $(muap)$ if, for every $\delta>0$, $X$ has an approximating
    sequence with complex (vs.~real) unconditional constant
    $1+\delta$. By \cite[Th.\ 3.8]{ck91} and \cite[Lemma 8.1]{gks93},
    this is the case if and only if there is an approximating sequence
    $(T_k)$ such that
    \begin{equation}\label{block:muap:def}
      \sup_{\epsilon\in\Sign}\|T_k+\epsilon (\Id-T_k)\|\longrightarrow1.
    \end{equation}
  \end{itemize}

\end{dfn}

$X$ has $(muap)$ if and
only if, for every given $\delta>0$, $X$ is isometric to a
1-complemented subspace of a space with a $(1+\delta)$-unconditional
finite-dimensional decomposition \cite[Cor.~IV.4]{gk97}. If $X$ has
$(muap)$, then, for any given $\delta>0$, $X$ is isometric to a
subspace of a space with a $(1+\delta)$-unconditional basis.

\begin{exa}
  The simplest example is the subspace in $\Sch^p$ of operators with an
  upper triangular matrix. In fact, if $I\subseteq\Row\times\Col$ is
  such that all columns $I\cap\Row\times\{c\}$ (vs.~all rows
  $I\cap\{r\}\times\Col$) are finite, then $\Sch^p_I$ admits a
  1-un\-con\-di\-tio\-nal finite-dimensional decomposition in the
  corresponding finitely supported idempotent Schur multipliers
  $\chi_{I\cap\Row\times\{c\}}$ (vs.~$\chi_{I\cap\{r\}\times\Col}$.)
\end{exa}

Our results on complete 1-un\-con\-di\-tio\-nal basic sequences yield the
following theorem.

\begin{thm}
  Let $1\le p\le\infty$. Let $\Row_r\subseteq\Row$, $r\in\N$, be pairwise
  disjoint and finite. Let $\Col_c\subseteq\Col$, $c\in\N$, be pairwise
  disjoint and finite. Let $J\subseteq\N\times\N$ and
  $I=\bigcup_{(r,c)\in J}\Row_r\times\Col_c$. Then the sequence of Schur
  multipliers $(\chi_{\Row_r\times\Col_c})_{(r,c)\in J}$ forms a
  complex 1-un\-con\-di\-tio\-nal finite-dimensional
  decomposition for $\Sch^p_I$ if and only if $J$ is a forest or $p$ is
  an even integer and $J$ contains no cycle of length $4,6,\dots,p$.
\end{thm}

We may always suppose that approximating sequences on spaces $\Sch^p_I$
are associated to Schur multipliers. More precisely, we have

\begin{prp}
  Let $1\le p\le\infty$ and $I\subseteq\Row\times\Col$.  Let $(T_n)$ be an
  approximating sequence on $\Sch^p_I$. Then there is a sequence of Schur
  multipliers $(\varphi_n)$ such that $(\Schur_{\varphi_n})$ is
  an approximating sequence on $\Sch^p_I$ and such that if 
  $(T_n)$ satisfies \eqref{block:muap:def}, then so does
  $(\Schur_{\varphi_n})$.
\end{prp}


\begin{proof}
  Let $\delta_n>0$ be such that $\delta_n\to0$.  As $T_n$ has finite
  rank, there is a finite $\Row_n\times\Col_n\subseteq\Row\times\Col$
  such that the projection $P_{\Row_n\times\Col_n}$ of $\Sch^p$
  onto $\Sch^p_{\Row_n\times\Col_n}$ defined by the Schur
  multiplier $\chi_{\Col_n}\otimes\chi_{\Row_n}$ satisfies
  $\|P_{\Row_n\times\Col_n}T_n-T_n\|<\delta_n$. 
  Let $\varphi_n$ be the Schur multiplier
  associated to $[T_n]_{\Row_n\times\Col_n}$. 
  With the notation of Eq.~\eqref{ave},
  \begin{equation*}
    M_{\varphi_n}(x) - x = \int_{\T^{\Row}}\mathrm{d}\eta\,
    \int_{\T^{\Col}}\mathrm{d}\zeta\,
    \Schur_{\zeta^*\otimes\eta^*}
    (P_{\Row_n\times\Col_n}T_n-\Id)(\Schur_{\zeta\otimes\eta}x).
  \end{equation*}
  As $P_{\Row_n\times\Col_n}T_n$ tends to the identity uniformly on
  compact sets, this shows that $M_{\varphi_n}$ is an approximating
  sequence.  As \[ \Schur_{\varphi_n} + \epsilon
  (\Id-\Schur_{\varphi_n}) = [P_{\Row_n\times\Col_n}T_n + \epsilon
  (\Id-P_{\Row_n\times\Col_n}T_n)]\ ,\] the norm of this operator is
  at most $\|T_n+\epsilon(\Id-T_n)\|+2\delta_n$.
\end{proof}

This proposition shows together with Prop.~\ref{pinf} the following results.

\begin{cor}
  Let $1\le p\le\infty$ and $I\subseteq\Row\times\Col$. 

  \begin{itemize}
  \item If $\Sch^p_I$ has $(muap)$, then  some
    sequence of Schur multipliers realises it.
  \item Let $J\subseteq I$. If $\Sch^p_I$ has $(muap)$, then so does $\Sch^p_J$.
  \item If $\Sch^\infty_I$ has $(muap)$, then so does $\Sch^p_I$.
  \end{itemize}

\end{cor}

Let us define the following asymptotic properties.

\begin{dfn}
    Let $1\le p\le\infty$, $I\subseteq\Row\times\Col$ and $\Sign=\T$
  (vs.~$\Sign=\{-1,1\}$.) 
  \begin{itemize}
  \item $\Sch^p_I$ is \emph{asymptotically unconditional} 
    if for every $x\in\Sch^p_I$ and for every bounded sequence
    $(y_n)$ in $\Sch^p_I$ such that each matrix coefficient of
    $y_n$ tends to 0
    \begin{equation*}
      \max_{\epsilon\in\Sign}\|x+\epsilon y_n\|_p -
      \min_{\epsilon\in\Sign}\|x+\epsilon y_n\|_p\longrightarrow0.
    \end{equation*}
  \item $I$ enjoys the property $(\mathscr{U})$ of \emph{block
    unconditionality} in $\Sch^p$ if for each $\delta>0$ and finite $F\subseteq I$, there
    is a finite $G\subseteq I$ such that 
    \begin{equation*}\forall\,x\in B_{\Sch^p_F}~\forall\,y\in
    B_{\Sch^p_{I\setminus G}}\quad 
      \max_{\epsilon\in\Sign}\|x+\epsilon y\|_p -
      \min_{\epsilon\in\Sign}\|x+\epsilon y\|_p < \delta.
    \end{equation*}
  \end{itemize}
\end{dfn}

The arguments of \cite[\S\,6.2]{ne98} show mutatis mutandis

\begin{thm}\label{muap}
  Let $1\le p\le\infty$, $I\subseteq\Row\times\Col$ and $\Sign=\T$
  \klammer{vs.~$\Sign=\{-1,1\}$.} Consider the following properties.
  
  \begin{enumerate}
  \item $\Sch^p_I$ is asymptotically unconditional.
  \item $I$ enjoys $(\mathscr{U})$ in $\Sch^p$.
  \item $\Sch^p_I$ has $(muap)$.
  \end{enumerate}

  Then $(c)\Rightarrow(a)\Leftrightarrow(b)$. If $1<p<\infty$, then $(b)\Leftrightarrow(c)$. If
  $p=1$, $\Sch^1_I$ has $(muap)$ if and only if $\Sch^1_I$ has $(uap)$ and
  $I$ enjoys $(\mathscr{U})$ in $\Sch^1$.
\end{thm}

The case $p=\infty$ is extreme in the sense that the following properties are
equivalent for $\Sch^\infty_I$: to be a dual space, to be reflexive, to
have a finite
cotype, not to contain $\cnaught$, because they are equivalent for $I$ not to contain
any sequence $(r_n,c_n)$ with $(r_n)$ and $(c_n)$ injective, that
is for $I$ to be
contained in the union of a finite set of lines and a finite set of columns,
so that $\Sch^\infty_I$ is isomorphic to $\ell^2_I$.

Let us now introduce the asymptotic property on $I$ that reflects the
combinatorics imposed by $(muap)$.

\begin{dfn}
  Let $I\subseteq\Row\times\Col$ and $k\ge1$. 

  \begin{itemize}
  \item $I$ enjoys property $\mathscr{J}_k$ if for every path
    $P=(c_0,r_0,\dots,c_j,r_j)$ of odd length $2j+1\le k$ in $I$ there
    is a finite set $\Row'\times\Col'$ such that $P$ cannot be completed
    with edges in $I\setminus\Row'\times\Col'$ to a cycle of length
    $2s\in\{4j+2,\dots,2k\}$.
  \item The \emph{asymptotic distance} $d_\infty(r,c)$ of
    $r\in\Row$ and $c\in\Col$ in $I$ is the supremum, over all finite
    rectangle sets $\Row'\times\Col'$, of the distance from $r$ to $c$
    in $I\setminus\Row'\times\Col'$.
  \end{itemize}

\end{dfn}

The asymptotic distance takes its values in $\{1,3,5,\dots,\infty\}$.
Note that $\mathscr{J}_1$ is true and that
$\mathscr{J}_k\Rightarrow\mathscr{J}_{k-1}$. This implication is strict: let $\Row,\Col$
be two copies of $\N$ and, given $j\ge1$, consider the union $I_j$ of
all paths $(\text{col}\,0,\allowbreak \text{row}\,{nj+1},\allowbreak
\text{col}\,{nj+1},\allowbreak \dots,\allowbreak
\text{row}\,{nj+j},\allowbreak \text{col}\,{nj+j},\allowbreak
\text{row}\,0)$ of length $2j+1$. Then $I_j$ contains no cycle of
length $2s\in\{4,\dots,4j\}$ and therefore enjoys $\mathscr{J}_{2j}$,
but fails $\mathscr{J}_{2j+1}$;
$I_j\cup\{(\text{row}\,0,\text{col}\,0)\}$ contains no cycle of length
$2s\in\{4,\dots,2j\}$ and thus enjoys $\mathscr{J}_{j}$, but
fails $\mathscr{J}_{j+1}$. In particular, the properties
$\mathscr{J}_k$, $k\ge2$, are not stable under union with a
singleton. 

Let us now explicit the relationship between $\mathscr{J}_k$ and $d_\infty$.

\begin{prp}
  Let $I\subseteq\Row\times\Col$ and $k\ge1$. 

  \begin{enumerate}
  \item $I$ enjoys $\mathscr{J}_k$ if and only if any two
    vertices $r\in\Row$ and $c\in\Col$ at distance $2j+1\le k$
    satisfy $d_\infty(r,c)\ge2k-2j+1$.
  \item If $d_\infty(r,c)\ge2k+1$ for all $(r,c)\in\Row\times\Col$,
    then $I$ enjoys $\mathscr{J}_k$.
  \item If $d_\infty(r,c)\le k$ for some $(r,c)\in\Row\times\Col$,
    then $I$ fails $\mathscr{J}_k$.
  \item $I$ enjoys $\mathscr{J}_k$ for every $k$ if and only if
    $d_\infty(r,c)=\infty$ for every $(r,c)\in\Row\times\Col$.
  \end{enumerate}

\end{prp}

\begin{proof}
  $(a)$ is but a reformulation of the definition of$\mathscr{J}_k$ and
  implies $(b)$. 

  $(d)$ is a consequence of $(b)$ and $(c)$.
  
  $(c)$. If $d_\infty(r,c)\le k$, then there is $0\le j\le(k-1)/2$
  such that there are infinitely many paths of length $2j+1$ from $c$
  to $r$: there is a path $(c,\allowbreak r_1,\allowbreak
  c_1,\dots,\allowbreak r_j,\allowbreak c_j,\allowbreak r)$ that can
  be completed with edges outside any given finite set to a cycle of
  length $4j+2\le2k$.
\end{proof}

\begin{thm}
  Let $I\subseteq\Row\times\Col$ and $1\le p\le\infty$. If $p$ is an even
  integer, then $\Sch^p_I$ has complex or real $(muap)$ if and only if
  $I$ enjoys $\mathscr{J}_{p/2}$. If $p=\infty$ or if $p$ is not an even
  integer, then $\Sch^p_I$ has real $(muap)$ only if
  $I$ enjoys $\mathscr{J}_k$ for every $k$.
\end{thm}

\begin{proof}
  Suppose that $I$ enjoys $(\mathscr{U})$ in $\Sch^p$ and fails $\mathscr{J}_k$.
  Then, for some $s\le k$, $I$ contains a sequence of cycles $(c_0,\allowbreak
  r_0,\dots,\allowbreak c_{j-1},\allowbreak r_{j-1},\allowbreak
  c_j^n,\allowbreak r_j^n,\dots,\allowbreak c_{s-1}^n,\allowbreak
  r_{s-1}^n)$
  with the property that $\|x-y\|_p\le(1+1/n)\|x+y\|_p$ for all $x$ with support
  in $\{(r_0,c_0),\allowbreak(r_0,c_1),\dots,\allowbreak
  (r_{j-2}, c_{j-1}),\allowbreak (r_{j-1}, c_{j-1})\}$ and all $y$ with
  support in $\{(r_{j-1}, c_j^n),\allowbreak
  (r_j^n, c_j^n),\allowbreak \dots,\allowbreak (r_{s-1}^n, c_{s-1}^n),\allowbreak
  (r_{s-1}^n, c_0)\}$. With the notation of Section~\ref{sec:cst}, this
  amounts to stating that the multiplier on
  $I=\{(i,i),(i,i+1)\}\subseteq\Z/s\Z\times\Z/s\Z$ given by
  $\epsilon_{rc}=1$ if $r,c\in\{0,\dots,j-1\}$ and $\epsilon_{rc}=-1$
  otherwise actually is an isometry on $\Sch^p_I$. As
  $\overline{\epsilon_{00}} \epsilon_{01} \dots
  \overline{\epsilon_{{s-1}\,{s-1}}}
  \epsilon_{{s-1}\,0}=(-1)^{2s-2j+1}=-1$, this implies by
  Prop.~\ref{cyno}$(e)$ that $p/2\in\{1,2,\dots,s-1\}$.
  
  Suppose that $I$ enjoys $\mathscr{J}_k$. We claim that for every finite
  $F\subseteq I$ there is a finite $G\subseteq I$ such that every
  closed walk $P\sim(\alpha,\beta)$ of length $2k$ in $I$ satisfies
  $\sum_{q\in I\setminus G}\beta_q-\alpha_q=0$. This signifies that given a
  closed walk $(v_0,\dots,v_{2k-1})$ and
  $0=a_0<b_0<\dots<a_m<b_m<a_{m+1}=2k$ such that
  $v_{a_i},\dots,v_{b_i-1}\in I\setminus G$ and
  $v_{b_i},\dots,v_{a_{i+1}-1}\in F$,
  \begin{equation*}
    \bigl\{i\in\{0,\dots,m\}:a_i,b_i~\text{even}\bigr\} =
    \bigl\{i\in\{0,\dots,m\}:a_i,b_i~\text{odd}\bigr\}.
  \end{equation*}
  Suppose that this
  is not true: then there is an $s\le k$, there are
  $0=a_0<b_0<\dots<a_m<b_m<2s$ and there are cycles
  $(v_{a_0}^n,\dots,\allowbreak v_{b_0-1}^n,\allowbreak
  v_{b_0},\dots,\allowbreak v_{a_1-1},\dots,\allowbreak
  v_{a_m}^n,\dots,\allowbreak v_{b_m-1}^n,\allowbreak
  v_{b_m},\dots,\allowbreak v_{2s-1})$ such that the $(v_i^n)_{n\ge0}$
  are injective sequences of vertices and $b_i-a_i$ is even for at
  least one index $i$: let us suppose so for $i=0$. If $b_0-a_0\ge
  s-1$, consider the path
  $P=(v_{b_0},\dots,\allowbreak v_{a_0-1},\allowbreak
  v_{a_0}^0,\dots,\allowbreak v_{b_m-1}^0,\allowbreak
  v_{b_m},\dots,\allowbreak v_{2s-1})$
  of odd length $2s-1-(b_0-a_0)$; if $b_0-a_0\le s-1$, consider
  the path $P=(v_{2s-1},\allowbreak v_{a_0}^0,\dots,\allowbreak
  v_{b_0-1}^0,\allowbreak v_{b_0})$ of odd
  length $b_0-a_0+1$. Then $P$ can be completed with vertices
  outside any given finite set to a cycle of length at most $2s$
  because $(v_{2s-1},v_{a_0}^n,\dots,\allowbreak
  v_{b_0-1}^n,\allowbreak v_{b_0})$ is a path of length $b_0-a_0+1$ in
  $I$ for every $n$. This proves that $I$ fails $\mathscr{J}_s$. 
  
  The claim shows that $I$ enjoys $(\mathscr{U})$ in $\Sch^p$ for $p=2k$. In
  fact, if $\tilde\epsilon\in\T^{F\cup(I\setminus G)}$ is defined by
  $\tilde\epsilon_q=1$ for $q\in F$ and
  $\tilde\epsilon_q=\epsilon\in\T$ for $q\in I\setminus G$, then, with
  the notation of Th.~\ref{sigma:ldc},
  \begin{equation*}
    \Phi_{F\cup(I\setminus G)}(\tilde\epsilon,a) =
    \sum_{(\alpha,\beta)\in\WR_{k}^{F\cup(I\setminus G)}}
    n_{\alpha\beta}\epsilon^{\sum_{q\in I\setminus G}
      \beta_q-\alpha_q}\overline{a}^\alpha a^\beta
  \end{equation*}
  does not depend on
  $\epsilon$, so that $\|x+\epsilon y\|_{2k}=\|x+y\|_{2k}$ if $x\in
  \Sch^{2k}_F$ and $y\in \Sch^{2k}_{I\setminus G}$, and $\Sch^{2k}_I$ has
  complex $(muap)$ by Th.~\ref{muap}$(b)\Rightarrow(c)$.
\end{proof}

\begin{rem}
  This theorem is a noncommutative analogue to \cite[Th.~7.5]{ne98}.
\end{rem}

\section{Examples}\label{ss11}

\noindent One of Varopoulos' motivations for the study of the projective tensor
product $\ell_\infty\projtens\ell_\infty$ are lacunary sets in a
locally compact abelian group. 

Let $\Gamma$ be a discrete abelian group and $\Lambda\subseteq\Gamma$.
Let us say that $\Lambda$ is \emph{$n$-independent} if every element
of $\Gamma$ admits at most one representation as the sum of $n$
terms in $\Lambda$, up to a permutation. For example, the geometric
sequence $\{j^k\}_{k\ge0}$ with $j\in\{2,3,\dots\}$ is $n$-independent
in $\Z$ if and only if $j\ge n$ \cite[\S\,3]{ne98}. If $\Lambda$ is
$n$-independent for all $n$, then $\Lambda$ is \emph{independent}.
Let
\[\Zeta_n=\{\zeta\in\Z^\Lambda:\sum_{\gamma\in\Lambda}\zeta_\gamma=0\
\text{and}\ \sum_{\gamma\in\Lambda}|\zeta_\gamma|\le2n]\] and
$\Zeta=\bigcup\Zeta_n$. Then $\Lambda$ is $n$-independent if and only
if, for every $\zeta\in\Zeta_n$,
\[\sum_{\gamma\in\Lambda}\zeta_\gamma\gamma=0\ \Rightarrow\ \zeta=0\]
and $\Lambda$ is independent if and only if this holds for every
$\zeta\in\Zeta$. 

Let us say that $\Lambda$ is \emph{$n$-independent modulo 2} if in
every representation of an element of $\Gamma$ as the sum of $n$
terms in $\Lambda$, each element of $\Lambda$ appears the same
number of times modulo 2. In other words, for every
$\zeta\in\Zeta_n$,
\[\sum_{\gamma\in\Lambda}\zeta_\gamma\gamma=0\quad
\Longrightarrow\quad \forall\,\gamma\in\Lambda\quad
\zeta_\gamma=0\pmod2\ ;\] $\Lambda$ is \emph{independent modulo
  2} if this holds for every $\zeta\in\Zeta$.
If $\Gamma$ contains no element of order 2, then one may always
suppose that at least one coefficient $\zeta_\gamma$ of a nontrivial
relation $\sum\zeta_\gamma\gamma=0$ is odd, so that these two latter
notions ``modulo 2'' coincide
with the two former ones.

Let $G=\hat\Gamma$, so that $\Gamma$ is the group of characters on
$G$. Then the computation presented in \cite[Prop.~2.5$(ii)$]{ne98}
for the case $\Gamma=\Z$ shows that $\Lambda$ is a complex
(vs.~real) 1-un\-con\-di\-tio\-nal basic sequence in
$\Ell^p(G)$ with $p\in2\N^*$
if and only if $\Lambda$ is $p/2$-independent (vs.~modulo 2).
Furthermore $\Lambda$ is a complex (vs.~real)
1-un\-con\-di\-tio\-nal basic sequence in $\Ell^{p}(G)$ with
$p\in(0,\infty]\setminus2\N^*$ if and only if $\Lambda$ is independent
(vs.~modulo 2). If $\Gamma$ contains no element of order
2, then a real 1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(G)$ is
also complex 1-un\-con\-di\-tio\-nal.  All these results hold also for the
complete counterparts to 1-un\-con\-di\-tio\-nal basic sequences.

Results on lacunary sets in a discrete abelian group transfer to lacunary
matrices in the following way, as in \cite[Th.~4.2]{va68}.

\begin{prp}\label{Gamma}
  Let $\Gamma$ be a discrete abelian group and $\Row$, $\Col$ be
  countable subsets of $\Gamma$. To every
  $\Lambda\subseteq\Row+\Col$ associate
  $I_\Lambda=\{(r,c)\in\Row\times\Col:r+c\in\Lambda\}$. Let
  $G=\hat\Gamma$. 

  \begin{enumerate}
  \item If $\Lambda$ is a complex 1-un\-con\-di\-tio\-nal basic sequence in
    $\Ell^4(G)$, then $I_\Lambda$ is a 1-un\-con\-di\-tio\-nal basic
    sequence in $\Sch^4$.
  \item Suppose that each element of $\Gamma$ admits at most one
    representation as the sum of an element of $\Row$ with an element
    of $\Col$. Then every $I\subseteq\Row\times\Col$ has the form
    $I=I_\Lambda$ with $\Lambda={\{r+c:(r,c)\in I\}}$. If $\Lambda$ is
    a real 1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(G)$, then $I_\Lambda$
    is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$.
  \item Let $p=2k$ be a positive even integer. Suppose that 
    $\Row\cap\Col=\emptyset$ and $\Row\cup\Col$ is $k$-independent modulo 2. 
    $I_\Lambda$ is a 1-un\-con\-di\-tio\-nal basic
    sequence in $\Sch^p$ if and only if $\Lambda$ is a real
    1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(G)$.
  \end{enumerate}
  
\end{prp}

\begin{proof}
  $(a)$. Let $P=(c,r,c',r')$ be a closed walk in $I_\Lambda$. Then
  $r+c$, $r'+c'$, $r+c'$ and $r'+c$ are in $\Lambda$ while
  $(r+c)+(r'+c')=(r+c')+(r'+c)$: if $\Lambda$ is
  2-independent, then $r+c\in\{r+c',r'+c\}$, so that $c=c'$ or
  $r=r'$ and $P$ is not a cycle. 
  
  $(b)$. For each $\gamma\in\Lambda$, let
  $q_\gamma=(r_\gamma,c_\gamma)$ be the unique element of $I$ such
  that $r_\gamma+c_\gamma=\gamma$. If $\Lambda$ is a real
  1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(G)$, then it is also a
  complete real 1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(G)$. Let
  $\varphi\in\{-1,1\}^{I_\Lambda}$, so that
  $\varphi_{q_\gamma}\in\{-1,1\}$ for all $\gamma\in\Lambda$. Then, as
  in Eq.~\eqref{tr1},
  \begin{multline*}
    \Bigl\| \sum_{q\in I_\Lambda} a_q\e_q
    \Bigr\|_{\Sch^p_{I_\Lambda}(\Sch^p)} = \Bigl\|
    \sum_{(r,c)\in I_\Lambda} r(g)c(g)a_{rc}\e_{rc}
    \Bigr\|_{\Sch^p_{I_\Lambda}(\Sch^p)}\\
    = \biggl\| \sum_{\gamma\in\Lambda}
    a_{q_\gamma}\e_{q_\gamma}\gamma(g)
    \biggr\|_{\Sch^p_{I_\Lambda}(\Sch^p)}
    = \biggl\| \sum_{\gamma\in\Lambda}
    a_{q_\gamma}\e_{q_\gamma}\gamma
    \biggr\|_{\Ell^p_\Lambda(G,\Sch^p(\Sch^p))}
  \end{multline*}
  so that as in Eq.~\eqref{tr2}, by complete real 1-un\-con\-di\-tio\-nality
  of $\Lambda$ in $\Ell^p(G)$,
  \begin{align*}
    \Bigl\| \sum_{q\in I_\Lambda} \varphi_qa_q\e_q
    \Bigr\|_{\Sch^p_{I_\Lambda}(\Sch^p)} = \biggl\|
    \sum_{\gamma\in\Lambda}
    \varphi_{q_\gamma}a_{q_\gamma}\e_{q_\gamma}\gamma
    \biggr\|_{\Ell^p_\Lambda(G,\Sch^p(\Sch^p))}= \Bigl\|
    \sum_{q\in I_\Lambda} a_q\e_q
    \Bigr\|_{\Sch^p_{I_\Lambda}(\Sch^p)}.
  \end{align*}

  $(c)$. Each element of $\Gamma$ admits at most one
    representation as the sum of an element of $\Row$ with an element
    of $\Col$, so that $(b)$ yields sufficiency. Suppose that $\Lambda$ is not a real
  1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(G)$ and let
  $\zeta\in\Zeta_k$ such that $\sum_{\gamma\in\Lambda}\zeta_\gamma\gamma=0$ and
  $J=\{(r,c)\in I_\Lambda:\zeta_{r+c}\ne0\pmod2\}$ is nonempty; $J$
  has at most $2k$ elements. 
  Let $P=(v_1,\dots,v_j)$ be a path in $J$ of maximal length. Then
  $\zeta_{v_{j-1}+v_j}$ is odd and
  $\sum\{\zeta_{v_j+v}:v_j+v\in\Lambda\}$ is even because it is the
  coefficient of $v_j$ in the relation $\sum_{\gamma\in\Lambda}\zeta_\gamma\gamma=0$ and
  $\Row\cup\Col$ is $k$-independent modulo 2. There is therefore $v_{j+1}$ distinct from
  $v_{j-1}$ such that $\zeta_{v_j+v_{j+1}}$ is odd. As $j$ is maximal
  and $\Row\cap\Col=\emptyset$, $v_{j+1}=v_{j+1-2i}$ for
  some $2\le i\le k$, so that $(v_{j+1-2i},\dots,v_j)$ is a cycle of
  length $2i$ in $J$: $I_\Lambda$ is not a 1-un\-con\-di\-tio\-nal basic
    sequence in $\Sch^p$. 
\end{proof}

Let $\Row$ and $\Col$ be any countable sets. Consider
$G=\{-1,1\}^\Col\times\{-1,1\}^\Row$. If we denote by
$\bigl((\epsilon_c)_{c\in\Col},\allowbreak(\epsilon'_r)_{r\in\Row}\bigr)$ a
generic point in $G$, then the set of Rademacher functions
$\{\epsilon_c\}_{c\in\Col}\cup\{\epsilon'_r\}_{r\in\Row}$ is a real
1-unconditional basic sequence in $\Cont(G)$, so that it is
independent modulo 2 in $\hat G$.
Similarly, the set of Steinhaus functions
$\{z_c\}_{c\in\Col}\cup\{z'_r\}_{r\in\Row}$ is independent in the dual
of $\T^\Col\times\T^\Row$.
This yields:

\begin{cor}\label{cor:S}
  Let $I\subseteq\Row\times\Col$ and $p\in(0,\infty]$. The following
  are equivalent:

  \begin{itemize}
  \item $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$.
  \item $\{\epsilon_c\epsilon'_r:(r,c)\in I\}$ is a real
    1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(G)$.
  \item $\{z_cz'_r:(r,c)\in I\}$ is a 
    1-un\-con\-di\-tio\-nal basic sequence in $\Ell^p(\T^\Col\times\T^\Row)$.
\end{itemize}

\end{cor}

\begin{rem}
  The isomorphic counterpart is also true: $I$ is a completely
  unconditional basic sequence in $\Sch^p$ (i.e., a
  complete $\sigma(p)$ set) if and only if
  $\{\epsilon_c\epsilon'_r:(r,c)\in I\}$ is a completely
  unconditio\-nal basic sequence in $\Ell^p(G)$ (a
  \emph{$\Lambda(p)_{\mathrm{cb}}$ set} in $\hat G$, see \cite{ha98}
  and \cite[\S\,8.1]{pi98},) if and only if $\{z_cz'_r:(r,c)\in I\}$
  is a completely unconditio\-nal basic sequence in
  $\Ell^p(\T^\Col\times\T^\Row)$.  This follows e.g.\ from the
  proof of Prop.~\ref{Gamma}$(b)$ and the iterated noncommutative
  Khinchin inequality \cite[Eq.\ (8.4.11)]{pi98}.
\end{rem}

Harcharras \cite{ha98} used Peller's discovery \cite{pe82a} of
the link between Fourier and Hankel Schur multipliers to produce
unconditional basic sequences in $\Sch^p$ that are unions of
antidiagonals in $\N\times\N$.  We have in our context the rather disappointing

\begin{prp}\label{hoory}
  Let $\Lambda\subseteq\N\subseteq\Z$ and
  $I=\{(r,c)\in\N\times\N:r+c\in\Lambda\}$.
  
  \begin{enumerate}
  \item $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^4$ if
    and only if $\{z^\lambda\}_{\lambda\in\Lambda}$ is a
    1-un\-con\-di\-tio\-nal basic sequence in $\Ell^4(\T)$. 
  \item If $\Lambda$ contains three elements $\lambda<\mu<\nu$ such that
    $\lambda+\mu\ge\nu$, then $I$ is not a
    1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$ if
    $p\in(0,\infty]\setminus\{2,4\}$.
    \item If $\Lambda=\{\lambda_k\}$ with $\lambda_{k+1}>2\lambda_k$
    for all $k$,
    then $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^p$
    for every $p$. 
  \end{enumerate}
  
\end{prp}

\begin{proof} $(a)$. Sufficiency follows from Prop.~\ref{Gamma}$(a)$ with
    $\Row=\Col=\N$. Conversely, if $\Lambda$ contains a solution to
  $\lambda+\mu=\lambda'+\mu'$ with $\lambda<\lambda'\le \mu'<\mu$, then $I$ contains the cycle
  $(\text{col}\,0,\text{row}\,\lambda,\text{col}\,\lambda'-\lambda,\text{row}\,\mu')$. 

$(b)$. Consider the cycle $(\text{col}\,0, \text{row}\,\lambda,
\text{col}\,\nu-\lambda,
\text{row}\,\mu-\nu+\lambda,
\text{col}\,\nu-\mu, \text{row}\,\mu)$. 

$(c)$. In fact, $I$ is a forest. Let $P=(c_1,r_1,\dots,c_k,r_k)$ be a
closed walk in $I$. We may suppose without loss of generality that
$r_1+c_2$ is a maximal element of
$\{r_1+c_1,r_1+c_2,\dots,r_k+c_k,r_k+c_1\}$. Then $r_1+c_1\le r_1+c_2$
and $r_2+c_2\le r_1+c_2$. One of these inequalities must be an
equality and $P$ is not a cycle: for otherwise $2(r_1+c_1)<r_1+c_2$
and $2(r_2+c_2)<r_1+c_2$ because $r_1+c_1,r_1+c_2,r_2+c_2\in\Lambda$,
so that $2(r_1+c_1+r_2+c_2)<2(r_1+c_2)$ and $c_1+r_2<0$.
\end{proof}

\begin{rem}
  Further computations yield the following result. If
  $\{z^\lambda\}_{\lambda\in\Lambda}$ is a 1-un\-con\-di\-tio\-nal
  basic sequence in $\Ell^6(\T)$ and if
  $\{\lambda<\mu<\nu\}\subseteq\Lambda\Rightarrow\lambda+\mu<\nu$,
  then $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in
  $\Sch^6$; the converse does not hold.
\end{rem}

Let us now give an overview of the known extremal bipartite graphs
without cycle of length $4,6,\dots,2k$ and their size.  Look up
\cite[Def.~I.3.1]{bjl99} for the definition of a Steiner system and
\cite[Def.~1.3.1]{va98} for the definition of a generalised
polygon. An  elementary example is given in the introduction with \eqref{steiner}. 

\begin{prp}\label{extr:prp}
  Let $2\le n\le m$, $I\subseteq\Row\times\Col$ with $\card\Col=n$ and $\card\Row=m$,
  and $e=\card I$.

  \begin{enumerate}
  \item If $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^4$,
    then 
    \begin{equation*}n\ge1 + \Bigl(\frac em-1\Bigr) + \Bigl(\frac
      em-1\Bigr)\Bigl(\frac en-1\Bigr),\end{equation*}
    that is
    $e^2-me-mn(n-1)\le0$. Equality holds if and only if $I$ is the
    incidence graph of a Steiner system $\mathrm S(2,e/m;n)$ on $n$ points and
    $m$ blocks.
  \item If $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^6$,
    then 
    \begin{equation*}n\ge1 + \Bigl(\frac em-1\Bigr) + \Bigl(\frac
      em-1\Bigr)\Bigl(\frac en-1\Bigr) + \Bigl(\frac
      em-1\Bigr)^2\Bigl(\frac en-1\Bigr),\end{equation*}
    that is
    $e^3-(m+n)e^2+2mne-m^2n^2\le0$. Equality holds if and only if $I$ is
    the incidence graph of the \emph{quadrangle} \klammer{the cycle of length 8} or
    of a generalised quadrangle with $n$ points and $m$ lines.
  \item If $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^{2k}$
    with $k\ge1$ an integer, then 
    \begin{equation}\label{hoo}
      n\ge\sum_{i=0}^k\Bigl(\frac em-1\Bigr)^{\lceil\frac i2\rceil}\Bigl(\frac
      en-1\Bigr)^{\lfloor\frac i2\rfloor}.
    \end{equation}
    Equality holds if $I$ is the incidence graph of the \emph{$(k+1)$-gon} \klammer{the
      cycle of length $2k+2$} or of a generalised $(k+1)$-gon with $n$
    points and $m$ lines.
  \end{enumerate}

\end{prp}

\begin{proof}
  By Theorem~\ref{sigma:THM}$(b)\Rightarrow(g)$, $I$ is a 1-un\-con\-di\-tio\-nal basic
  sequence in $\Sch^{2k}$, with $k\ge1$ an integer, if and only if $I$ is a
  graph of girth $2k+2$ in the sense of \cite{ho02}. Therefore $(a)$ and $(b)$ are shown in
  \cite[Prop.~4, Th.~8, Rem.~10]{ne01}. Inequality \eqref{hoo} is
  \cite[Eq.~(1)]{ho02} and the sufficient condition for equality follows from 
  \cite[Lemma 1.5.4]{va98}. 
\end{proof}

Consult \cite[Tables~A1.1, A5.1]{bjl99} for examples of Steiner
systems and \cite[Table~2.1]{va98} for examples of generalised
polygons. In both cases, the corresponding incidence graph is biregular: every
vertex in $\Row$ has same degree $s+1$ and every vertex in $\Col$ has same
degree $t+1$. Arbitrarily large generalised $(k+1)$-gons exist only
if $2k\in\{4,6,10,14\}$ \cite[Lemma 1.7.1]{va98}; for
$2k\in\{6,10,14\}$, it follows from \cite[Lemma 1.5.4]{va98} that 
\begin{equation*}
  n=(s+1)\frac{(st)^{(k+1)/2}-1}{st-1},~
  m=(t+1)\frac{(st)^{(k+1)/2}-1}{st-1}.
\end{equation*} 

\begin{rem}\label{extr:rem}
  Let $I\subseteq\Row\times\Col$ with $\card\Col=\card\Row=n$. Inequality~\eqref{hoo} shows
  that if $I$ is a 1-un\-con\-di\-tio\-nal basic sequence in $\Sch^{2k}$, then
  $\card I\le n^{1+1/k}+(s-1)n/s$. If $p\notin\{4,6,10\}$, the existence
  of 1-un\-con\-di\-tio\-nal basic sequences in $\Sch^{2k}$ such that
  $\card I\succcurlyeq n^{1+1/k}$ is in fact an important open problem in graph
  theory: extremal graphs cannot correspond to generalised polygons
  and necessarily have less structure.
\end{rem}

\begin{acknowledgements}
  I would like to thank \'Eric Ricard for his comments and suggestions.
\end{acknowledgements}


\def\cprime{$'$}\def\bibmath{disponible \`a la biblioth\`eque de
  math\'ematiques}\def\bibirem{disponible \`a la biblioth\`eque de
  l'IREM}\def\BUL{disponible \`a la BU Lettres}\def\BUS{disponible \`a la BU
  Sciences}\ifx\iflanguage\undefined\def\iflanguage#1#2#3{#3}\fi
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\end{thebibliography}


\noindent
\emph{Keywords.} Schatten-von-Neumann class, Schur product, graph with a
given girth, 1-unconditional basic sequence, metric unconditional
approximation property, V-Sidon set, lacunary set. 

\bigskip

\noindent 
2000 \emph{Mathematics Subject Classification.} 47B10, 46B15, 46B04,
43A46, 05C38, 46B28.


\end{document}
