# Cycles and 1-unconditional matrices

Stefan Neuwirth

► To cite this version:

Stefan Neuwirth. Cycles and 1-unconditional matrices. Proceedings of the London Mathematical Society, 2006, 93 (3), pp.761-790. 10.1017/S0024611506015899. hal-00473583

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# Cycles and 1-unconditional matrices

Stefan Neuwirth

## Abstract

We characterise the 1-unconditional subsets $(e_{rc})_{(r,c)\in I}$ of the set of elementary matrices in the Schatten-von-Neumann class $S^p$. 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 $S_I^p$ spanned by $(e_{rc})_{(r,c)\in I}$ in $S^p$.

## Résumé en français

Je caractérise les sous-suites 1-inconditionnelles $(e_{rc})_{(r,c)\in I}$ de la suite des matrices élémentaires dans la classe de Schatten-von-Neumann $S^p$. L'ensemble de couples $I$ doit être l'ensemble des arêtes 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 réel positif qui n'est pas un entier pair. Dans ce dernier cas, $I$ est même un ensemble de Varopoulos de V-interpolation de constante 1. J'étudie aussi la propriété d'approximation inconditionnelle métrique pour le sous-espace vectoriel fermé $S_I^p$ engendré par $(e_{rc})_{(r,c)\in I}$ dans $S^p$.

## 1 Introduction

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

*Question 1.1.* Which matrix coefficients of an operator $x \in S^p$ must vanish so that the norm of $x$ does not depend on the argument, or on the sign, of the remaining nonzero matrix coefficients?

Let $C$ be the set of columns and $R$ be the set of rows for coordinates in the matrix. Let $I \subseteq R \times C$ be the set of matrix coordinates of the nonzero matrix coefficients of $x$ (the *pattern*.) *Question 1.1* describes the notion of a complex, or real, 1-unconditional basic sequence $(e_{rc})_{(r,c)\in I}$ of elementary matrices in $S^p$ (see *Definition 4.1*.)

By a convexity argument, *Question 1.1* is equivalent to the following question on Schur multiplication.

*Question 1.2.* Which matrix coefficients of an operator $x \in S^p$ must vanish so that for all matrices $\varphi$ of complex, or real, numbers

$$ \|\varphi * x\| \le \sup |\varphi_{rc}| \|x\|, $$

where $\varphi * x$ is the Schur (or Hadamard or entrywise) product defined by

$$ (\varphi * x)_{rc} = \varphi_{rc} x_{rc}? $$

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 $\ell_C^\infty \hat{\otimes} \ell_R^\infty$: this norm is equivalent to the supremum of the norm of those elements of $\ell_C^\infty \hat{\otimes} \ell_R^\infty$ whose coefficient matrices are finite submatrices of $\varphi$. In the framework of tensor algebras over discrete spaces, *Question 1.2* turns out to describe as well the isometric counterpart to Varopoulos' V-Sidon sets as well as to his sets of V-interpolation. The following isometric question has however a different answer.
**Question 1.3.** Which coefficients of a tensor $u \in \ell_C^\infty \hat{\otimes} \ell_R^\infty$ must vanish so that the norm of $u$ is the maximal modulus of its coefficients?

In our answer to Question 1.2, $S^p$ and Schur multiplication are treated as a noncommutative analogue to $L^p$ and convolution. The main step is a careful study of the Schatten-von-Neumann norm $\|x\| = (\text{tr}(x^*x)^{p/2})^{1/p}$ 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 tuples $(v_1, v_2, \dots, v_p)$ satisfying $(v_{2i-1}, v_{2i}), (v_{2i+1}, v_{2i}) \in I$, where $v_{p+1} = v_1$: see the computation in Eq. (10). These are best understood as closed walks of length $p$ on the bipartite graph $G$ canonically associated to $I$: its vertex classes are $C$ and $R$ 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 1.1 and 1.2.

**Theorem 1.4.** 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-unconditional basic sequence in $S^p$, then the graph $G$ associated to $I$ contains no cycle. In this case, $I$ is even a set of V-interpolation with constant 1: every sequence $\varphi \in \ell_I^\infty$ may be interpolated by a tensor $u \in \ell_C^\infty \hat{\otimes} \ell_R^\infty$ such that $\|u\| = \|\varphi\|$.

**Theorem 1.5.** Let $p \in \{2, 4, 6, \dots\}$. The sequence $(e_{rc})_{(r,c)\in I}$ is a complex, or real, 1-unconditional basic sequence in $S^p$ if and only if $G$ contains no cycle of length 4, 6, ..., $p$.

These theorems hold also for the complete counterparts to 1-unconditional basic sequences in the sense of Def. 4.1(c).

In particular, if we denote by $U_p$ the property that $(e_{rc})_{(r,c)\in I}$ is a 1-unconditional basic sequence in $S^p$, then we obtain the following hierarchy:

$$U_p \text{ for a } p \in (0, \infty] \setminus \{2, 4, 6, \dots\} \Rightarrow \dots \Rightarrow U_{2n+2} \Rightarrow U_{2n} \Rightarrow \dots \Rightarrow U_2.$$

If $C$ and $R$ 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 $C$ and $R$ become large. Proposition 11.6 and Remark 11.7 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:

* Suppose that $p$ is not an even integer. If a pattern $I$ contains the vertices of such a polygon, then there is an operator $x \in S^p$ whose matrix coefficients vanish outside $I$ and whose norm depends on the sign of its matrix coefficients. This condition is also necessary.

* 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 $S^p$ 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.

An elementary example is given by the set

$$I = \{(r, c) \in \mathbb{Z}/7\mathbb{Z} \times \mathbb{Z}/7\mathbb{Z} : r + c \in \{0, 1, 3\}\}. \quad (1)$$

The associated bipartite graph is known as the Heawood graph (Fig. 1:) it is the incidence graph of the Fano plane (the finite projective plane PG(2, 2),) which is the smallest generalised triangle, and corresponds to the Steiner system S(2, 3; 7). It contains no cycle of length 4, but every pair of vertices is contained in a cycle of length 6.
<figure><img src="image_2.png" /><figcaption>Figure 1: The Heawood graph</figcaption></figure>

Thus the *-trace norm of every matrix with pattern

$$ \begin{pmatrix} 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ * & * & 0 & * & 0 & 0 & 0 \\ * & 0 & * & 0 & 0 & 0 & * \\ 0 & * & 0 & 0 & 0 & * & * \\ * & 0 & 0 & 0 & * & * & 0 \\ 0 & 0 & 0 & * & * & 0 & * \\ 0 & * & * & 0 & * & 0 & 0 \end{pmatrix} $$

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-unconditional basis of the space $S_I^p$ it spans in $S^p$. If this is not the case, $S_I^p$ might still admit some other 1-unconditional basis. This leads to the following more general question.

**Question 1.6.** For which sets $I$ does $S_I^p$ admit some kind of almost 1-unconditional finite dimensional expansion of the identity?

The metric unconditional approximation property (muap) provides a formal definition for the object of Question 1.6: see Def. 10.1. We obtain the following results.

**Theorem 1.7.** Let $p \in [1, \infty] \setminus \{2, 4, 6, \dots\}$. If $S_I^p$ 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 $G$.

**Theorem 1.8.** Let $p \in \{2, 4, 6, \dots\}$. The space $S_I^p$ 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 $p - 2j + 1$.

We now turn to a detailed description of this article. In Section 2, we provide tools for the computation of Schur multiplier norms. Section 3 characterises idempotent Schur multipliers and 0, 1-tensors in $\ell_C^\infty \hat{\otimes} \ell_R^\infty$ of norm 1. In Section 4, we define the complex and real unconditional constants of basic sequences of elementary matrices and show that they are not equal in general. Section 5 looks back on Varopoulos' results about tensor algebras over discrete spaces. Section 6 puts the connection between *-trace norm and closed walks of length * in the concrete form of closed walk relations. In Section 7, 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 8 is dedicated to a proof of Th. 1.4 and an answer to Question 1.3. Section 9 establishes Th. 1.5. In Section 10, we study the metric unconditional approximation property for spaces $S_I^p$. The final section provides four kinds of examples: sets obtained by a transfer of *-independent subsets of a discrete abelian group, Hankel sets, Steiner systems and Tsits' generalised polygons.

**Terminology.** $C$ is the set of *columns* and $R$ is the set of *rows*, both finite or countable and if necessary indexed by natural numbers. $V$, the set of *vertices*, is their disjoint union $C \amalg R$: 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 edge on $V$ is a pair $\{v, w\} \subseteq V$. A graph on $V$ is given by a set of edges $E$. A bipartite graph on $V$ with vertex classes $C$ and $R$ has only edges $\{r, c\}$ such that $c \in C$ and $r \in R$ and may therefore be given alternatively by the set of couples $I = \{(r, c) \in R \times C : \{r, c\} \in E\}$: this will be our custom throughout the article. A bipartite graph on $V$ is complete if its set of couples $I$ is the whole of $R \times C$. Two graphs are disjoint if so are the sets of vertices of their edges. $I$ is a column section if $(r, c)$, $(r', c) \in I \Rightarrow r = r'$, and a row section if $(r, c)$, $(r, c') \in I \Rightarrow c = c'$.

A walk of length $s \ge 0$ in a graph is a sequence $(v_0, \dots, v_s)$ of $s+1$ vertices such that $\{v_0, v_1\}$, \dots, $\{v_{s-1}, v_s\}$ are edges of the graph. A walk is a path if its vertices are pairwise distinct. The 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 closed walk of length $p \ge 0$ 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 cycle if its vertices are pairwise distinct. We take the convention that if a closed walk in a bipartite graph on $V = C \amalg R$ is nonempty, then its first vertex is a column vertex: $v_1 \in C$. 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 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 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 $c \cdots \cdots v$ index $v$. Let $W$ be the set of all words thus formed. Then

* $\emptyset \in W$ and every beginning of a word in $W$ is also in $W$: if $w \in W \setminus \{\emptyset\}$, then $w$ is the concatenation $w' \hat{v}$ of a word $w' \in W$ with a letter $v$;

* words of even length index row vertices;

* words of odd length index column vertices;

* a pair of vertices is an edge exactly if their indices have the form $w$ and $w \hat{v}$, where $w$ is a word and $v$ is a letter.

A forest is a union of pairwise disjoint trees; equivalently, it is a cycle free graph.

**Notation.** Let $\mathbb{T} = \{z \in \mathbb{C} : |z| = 1\}$.

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 $I = R \times C$ and $q = (r, c)$. Then $e_q = e_{rc}$ is the elementary matrix identified with the operator from $\ell_C^2$ to $\ell_R^2$ that maps $e_c$ on $e_r$ and all other basis vectors on 0. The matrix coefficient at coordinate $q$ of an operator $x$ from $\ell_C^2$ to $\ell_R^2$ is $x_q = \text{tr} e_q^* x$ and its matrix representation is $(x_q)_{q \in R \times C} = \sum_{q \in R \times C} x_q e_q$. The support of $x$ is $\{q \in R \times C : x_q \neq 0\}$.

The Schatten-von-Neumann class $S^p$, $0 < p < \infty$, is the space of those compact operators $x$ from $\ell_C^2$ to $\ell_R^2$ such that $\|x\|_p^p = \text{tr}|x|^p = \text{tr}(x^*x)^{p/2} < \infty$. $S^\infty$ is the space of compact operators with the operator norm. $S^p$ is a quasi-normed space, and a Banach space if $p \ge 1$. Let $(R_n \times C_n)_{n \ge 0}$ be a sequence of finite sets that tends to $R \times C$. Then the sequence of operators $P_n: x \mapsto \sum_{q \in R_n \times C_n} x_q e_q$ tends pointwise to the identity on $S^p$ if $p \ge 1$.

For $I \subseteq R \times C$, the entry space $S_I^p$ is the subspace of those $x \in S^p$ whose support is a subset of $I$. $S_I^p$ is also the closed subspace of $S^p$ spanned by $(e_q)_{q \in I}$.

The $S^p$-valued Schatten-von-Neumann class $S^p(S^p)$ is the space of those compact operators $x$ from $\ell_C^2$ to $\ell_R^2(S^p)$ such that $\|x\|_p^p = \text{tr}(\text{tr}|x|^p) < \infty$, where the inner trace is the $S^p$-valued analogue of the usual trace: such operators have an $S^p$-valued matrix representation and their support is defined as in the scalar case. An element $x \in S^p(S^p)$ can also be considered as a compact operator from $\ell_C^2(\ell_2) = \ell_2 \otimes_2 \ell_C^2$ to $\ell_R^2(\ell_2) = \ell_2 \otimes_2 \ell_R^2$ such that $\|x\|_p^p = \text{tr} \otimes \text{tr}|x|^p < \infty$; the matrix coefficient of $x$ at $q$ is then $x_q = (\text{Id}_{S^p} \otimes \text{tr})((\text{Id}_{\ell_2} \otimes e_q^*)x)$ and its matrix representation is $\sum_{q \in R \times C} x_q \otimes e_q$. The entry space $S_I^p(S^p)$ is defined in the same way as $S_I^p$.

A relative Schur multiplier on $S_I^p$ is a sequence $\varphi = (\varphi_q)_{q \in I} \in \mathbb{C}^I$ such that the associated Schur multiplication operator $M_\varphi$ defined by $e_q \mapsto \varphi_q e_q$ for $q \in I$ is bounded on $S_I^p$. The Schur multiplier $\varphi$ is furthermore completely bounded (c.b. for short) on $S_I^p$ if $\text{Id}_{S^p} \otimes M_\varphi$, the operator defined by $x_q e_q \mapsto \varphi_q x_q e_q$ for $x_q \in S^p$ and $q \in I$, is bounded on $S_I^p(S^p)$ (see [21, Lemma 1.7].) The norm of
$\varphi$ is the norm of $M_\varphi$ and its complete norm is the norm of $\mathrm{Id}_{S^p} \otimes M_\varphi$. This norm is the supremum of the norm of its restrictions to finite rectangle sets $R' \times C'$. Note that $\varphi$ is a Schur multiplier on $S^\infty$ if and only if, for every bounded operator $x: \ell_C^2 \to \ell_R^2$, $(\varphi_q x_q)$ is the matrix representation of a bounded operator; also $\varphi$ is automatically c.b. on $S^\infty$ [22, Th. 5.1]. We used [21, 22] as a reference.

Let $G$ be a compact abelian group endowed with its normalised Haar measure. Let $\Gamma = \hat{\Gamma}$ be the dual group of characters on $G$. The spectrum of an integrable function $f$ on $G$ is $\{\gamma \in \Gamma : \hat{f}(\gamma) \neq 0\}$. Let $\Lambda \subseteq \Gamma$. If $X$ is a space of integrable functions on $G$, then $X_\Lambda$ is the translation invariant subspace of those $f \in X$ whose spectrum is a subset of $\Lambda$.

Let $X$ be the space of continuous functions $C(G)$ or the Lebesgue space $L^p(G)$ with $0 < p < \infty$. Then $X_\Lambda$ is also the closed subspace of $X$ spanned by $\Lambda$. A relative Fourier multiplier on $X_\Lambda$ is a sequence $\mu = (\mu_\gamma)_{\gamma \in \Lambda} \in \mathbb{C}^\Lambda$ such that the associated convolution operator $M_\mu$ 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 $\mathrm{Id}_{S^p} \otimes M_\mu$, the operator defined by $a_\gamma \gamma \mapsto \mu_\gamma a_\gamma \gamma$ for $a_\gamma \in S^p$ and $\gamma \in \Lambda$, is bounded on the $S^p$-valued space $X_\Lambda(S^p)$ (where $p = \infty$ if $X = C(G)$.) The norm of $\mu$ is the norm of $M_\mu$ and its complete norm is the norm of $\mathrm{Id}_{S^p} \otimes M_\mu$. Note that $\mu$ is a Fourier multiplier on $C_\Lambda(G)$ if and only if, for every $f \in L_\Lambda^\infty(G)$, $\sum \mu_\gamma \hat{f}(\gamma) \gamma$ is the Fourier series of an element of $L_\Lambda^\infty(G)$: $\mu$ is a relative Fourier multiplier on $L^\infty(G)$; also $\mu$ is automatically c.b. on $C_\Lambda(G)$ [22, Cor. 3.18].

Let $X, Y$ be Banach spaces and $u \in X \otimes Y$. Its projective tensor norm is

$$ \|u\|_{X \hat{\otimes} Y} = \inf \left\{ \sum_{j=1}^{n} \|x_j\| \|y_j\| : u = \sum_{j=1}^{n} x_j \otimes y_j \right\} $$

and $X \hat{\otimes} Y$ is the completion of $X \otimes Y$ with respect to this norm. Note that $\ell_\infty^n \hat{\otimes} \ell_\infty^m \subset c_0 \hat{\otimes} c_0$ because $\ell_\infty^n$ and $\ell_\infty^m$ are 1-complemented in $c_0$, and that $c_0 \hat{\otimes} c_0 \subset \ell_\infty \hat{\otimes} \ell_\infty$ because $\ell_\infty$ is the bidual of $c_0$.

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 injective tensor norm of $u$ is

$$ \|u\|_{X \hat{\otimes} Y} = \sup_{(\xi, \eta) \in B_{X^*} \times B_{Y^*}} |\langle \xi \otimes \eta, u \rangle| $$

and $X \hat{\otimes} Y$ is the completion of $X \otimes Y$ with respect to this norm.

If $X$ and $Y$ are both finite dimensional, then

$$ (X \hat{\otimes} Y)^* = X^* \hat{\otimes} Y^* \quad \text{and} \quad (X \hat{\otimes} Y)^* = X^* \hat{\otimes} Y^* $$

Further $(c_0 \hat{\otimes} c_0)^* = \ell_1 \hat{\otimes} \ell_1$: in fact, $(c_0 \hat{\otimes} c_0)^*$ may be identified with the space of bounded operators from $c_0$ to $\ell_1$ and $\ell_1 \hat{\otimes} \ell_1$ may be identified with the closure of finite rank operators in that space, and they are the same because every bounded operator from $c_0$ to $\ell_1$ is compact and $\ell_1$ has the approximation property.

If $X$ is a sequence space on $C$ and $Y$ is a sequence space on $R$, then the coefficient of the tensor $u$ at $(r, c)$ is $\langle e_c \otimes e_r, u \rangle$. Its support is the set of coordinates $(r, c)$ of its nonvanishing coefficients. One may use [26] as a reference.

## 2 Relative Schur multipliers

The following proposition is a straightforward consequence of [17].

**Proposition 2.1.** Let $I \subseteq R \times C$ and $\varphi$ be a Schur multiplier on $S_I^\infty$ with norm $D$. Then $\varphi$ is also a c.b. Schur multiplier on $S_I^p$ for every $p \in (0, \infty]$, with complete norm bounded by $D$.

*Proof.* We may assume that $D = 1$. Let $R' \times C'$ be any finite subset of $R \times C$. By [17, Th. 3.2], 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 $(r, c) \in I \cap R' \times C'$. If we define $W: \ell_{C'}^2 \to \ell_{C'}^2(H)$ and $V: \ell_{R'}^2 \to \ell_{R'}^2(H)$ by $W\zeta = (\zeta_c w_c)_{c \in C'}$
and $V\eta = (\eta_r v_r)_{r \in R'}$, then $V$ and $W$ have norm at most 1, and the proposition follows from the factorisation

$$M_{\phi}x = V^*(x \otimes \mathrm{Id}_H)W$$

for every $x$ with support in $I \cap R' \times C'$.

<figure><img src="image_3.png" /></figure>

**Remark 2.2.** Éric Ricard showed us an elementary proof that a Schur multiplier on $S_I^\infty$ is automatically c.b., included here by his kind permission. A Schur multiplier $\phi$ is bounded on $S_I^\infty$ by a constant $D$ if and only if

$$\forall \xi \in B_{S_I^\infty} \quad \forall \eta \in B_{\ell_R^2} \quad \forall \zeta \in B_{\ell_C^2} \quad \left| \sum_{(r,c) \in I} \eta_r \phi_{rc} \xi_{rc} \zeta_c \right| \le D. \qquad (2)$$

It is furthermore completely bounded on $S_I^\infty$ by $D$ if

$$\forall x \in B_{S_I^\infty}(S^\infty) \quad \forall y \in B_{\ell_R^2}(\ell_2) \quad \forall z \in B_{\ell_C^2}(\ell_2) \quad \left| \sum_{(r,c) \in I} \phi_{rc} \langle y_r, x_{rc} z_c \rangle \right| \le D. \qquad (3)$$

Suppose that $x, y, z$ are as quantified in Ineq. (3). Let

$$\xi_{rc} = \langle y_r / \|y_r\|, x_{rc} z_c / \|z_c\| \rangle, \quad \eta_r = \|y_r\|_{\ell_2} \quad \text{and} \quad \zeta_c = \|z_c\|_{\ell_2}.$$

Then $\|\eta\|_{\ell_R^2}, \|\zeta\|_{\ell_C^2} \le 1$ and

$$\|\xi\| = \sup \left\{ \left| \sum_{(r,c) \in I} \langle \alpha_r y_r / \|y_r\|_{\ell_2}, x_{rc} \beta_c z_c / \|z_c\|_{\ell_2} \rangle \right| : \alpha \in B_{\ell_R^2}, \beta \in B_{\ell_C^2} \right\} \\ \le \|x\| \sup_{\alpha \in B_{\ell_R^2}} \| (\alpha_r y_r / \|y_r\|_{\ell_2}) \|_{\ell_R^2}(\ell_2) \sup_{\beta \in B_{\ell_C^2}} \| (\beta_c z_c / \|z_c\|_{\ell_2}) \|_{\ell_C^2}(\ell_2) \le 1,$$

so that Ineq. (2) implies Ineq. (3).

The fact that the canonical basis of an $\ell^2$ space is 1-unconditional yields that Schatten-von-Neumann norms are *matrix unconditional* in the terminology of [27]:

$$\forall \zeta \in \mathbb{T}^C \quad \forall \eta \in \mathbb{T}^R \quad \left\| \sum_{(r,c) \in R \times C} \zeta_c \eta_r a_{rc} e_{rc} \right\|_p = \left\| \sum_{(r,c) \in R \times C} a_{rc} e_{rc} \right\|_p \qquad (4)$$

for every finitely supported sequence of complex or $S^p$-valued coefficients $a_{rc}$. Let $\zeta \otimes \eta$ denote the elementary Schur multiplier $(\zeta_c \eta_r)_{(r,c) \in R \times C}$. Equation (4) shows that if $\zeta \in \mathbb{T}^C$ and $\eta \in \mathbb{T}^R$, then $M_{\zeta \otimes \eta}$ is an isometry on every $S^p$. This yields that if $\zeta \in \ell_C^\infty$, $\eta \in \ell_R^\infty$, then the complete norm of $M_{\zeta \otimes \eta}$ is $\|\zeta\|_{\ell_R^\infty} \|\eta\|_{\ell_C^\infty}$ on every $S^p$.

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

**Definition 2.3.** Let $T: S_J^p \to S_I^p$ be an operator. Let $R' \times C'$ be a finite subset of $R \times C$ and let $P_{R' \times C'}$ be the contractive projection onto $S_{R' \times C'}^p$ defined by the Schur multiplier $\chi_{C'} \otimes \chi_{R'}$. Then the *average of T with respect to R' \times C'* is given by

$$[T]_{R' \times C'}(x) = \int_{\mathbb{T}^R} d\eta \int_{\mathbb{T}^C} d\zeta M_{\zeta^* \otimes \eta^*} P_{R' \times C'} T(M_{\zeta \otimes \eta} x), \qquad (5)$$

where $\zeta^* = (\bar{\zeta}_c)_{c \in C}$ and $\eta^* = (\bar{\eta}_r)_{r \in R}$.

**Proposition 2.4.** Let $T: S_J^p \to S_I^p$ be an operator and $R' \times C'$ a finite subset of $R \times C$. Then $[T]_{R' \times C'}$ is a Schur multiplication operator from $S_J^p$ to $S_{I \cap R' \times C'}^p$ such that $\|[T]_{R' \times C'}\| \le \|T\|$. In fact, $[T]_{R' \times C'} = M_{\phi_{R' \times C'}}$ with

$$\varphi_{rc}^{R' \times C'} = \begin{cases} \mathrm{tr} e_{rc}^* T(e_{rc}) & \text{if } (r, c) \in J \cap R' \times C' \\ 0 & \text{if } (r, c) \in J \setminus R' \times C'. \end{cases}$$

If $T$ is a projection onto $S_I^p$, then $\varphi^{R' \times C'} = \chi_{I \cap R' \times C'}$, so that $[T]_{R' \times C'}$ is a projection onto $S_{I \cap R' \times C'}^p$. Let $\varphi = (\mathrm{tr} e_q^* T(e_q))_{q \in J}$. Then $\|M_{\varphi}\| \le \|T\|$ and we define the average of $T$ by $[T] = M_{\varphi}$.
Proof. Formula (5) shows that $||[T]_{R' \times C'}(x)|| \le ||T|| ||x||$. We have

$$
\begin{align*}
[T]_{R' \times C'}(e_{rc}) &= \int_{\mathbb{T}^R} d\eta \int_{\mathbb{T}^C} d\zeta M_{\zeta^* \otimes \eta^*} P_{R' \times C'} T(\zeta_c \eta_r e_{rc}) \\
&= \int_{\mathbb{T}^R} d\eta \int_{\mathbb{T}^C} d\zeta \zeta_c \eta_r M_{\zeta^* \otimes \eta^*} \sum_{(r',c') \in R' \times C'} \mathrm{tr}(e_{r'c'}^* T(e_{rc})) e_{r'c'} \\
&= \sum_{(r',c') \in R' \times C'} \int_{\mathbb{T}^R} d\eta \int_{\mathbb{T}^C} d\zeta \zeta_c \eta_r \mathrm{tr}(e_{r'c'}^* T(e_{rc})) \zeta_{c'}^{-1} \eta_{r'}^{-1} e_{r'c'} = \varphi_{rc}^{R' \times C'} e_{rc}.
\end{align*}
$$

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 $S_I^p$ and $||M_\varphi|| \le ||T||$. If $T$ is a projection onto $S_I^p$, note that $\mathrm{tr} e_{rc}^* T(e_{rc}) = \chi_I(r, c)$. $\square$

The following proposition relates Fourier multipliers to Herz-Schur multipliers in the fashion of [22, Th. 6.4] and will be very useful in the exact computation of the norm of certain relative Schur multipliers.

**Proposition 2.5.** Let $\Gamma$ be a countable discrete abelian group and $\Lambda \subseteq \Gamma$. Let $R$ and $C$ be two copies of $\Gamma$ and consider $I = \{(r, c) \in R \times C : r - c \in \Lambda\}$. Let $\varphi \in \mathbb{C}^I$ such that there is $\mu \in \mathbb{C}^\Lambda$ 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 $p \in (0, \infty]$.

(a) The complete norm of the relative Schur multiplier $\varphi$ on $S_I^p$ is bounded by the complete norm of the relative Fourier multiplier $\mu$ on $L_\Lambda^p(G)$.

(b) Suppose that $\Gamma$ is finite. The norm of the relative Fourier multiplier $\mu$ on $L_\Lambda^p(G)$ is bounded by the norm of the relative Schur multiplier $\varphi$ on $S_I^p$. The same holds for complete norms.

*Remark 2.6.* Part (b) is just an abstract counterpart to [20, Chapter 6, Lemma 3.8], where the case of the finite cyclic group $\Gamma = \mathbb{Z}/n\mathbb{Z}$ is treated.

*Proof.* (a) is [21, Lemma 8.1.4]: for all $a_q \in S^p$, of which only a finite number are nonzero, and all $g \in G$, we have by matrix unconditionality (Eq. (4))

$$
\begin{align}
\left\| \sum_{q \in I} a_q e_q \right\|_{S_I^p(S^p)} &= \left\| \sum_{(r,c) \in I} r(g)c(g)^{-1} a_{rc} e_{rc} \right\|_{S_I^p(S^p)} \nonumber \\
&= \left\| \sum_{\gamma \in \Lambda} \left( \sum_{(r,c) \in I \atop r-c=\gamma} a_{rc} e_{rc} \right) \gamma(g) \right\|_{S_I^p(S^p)} = \left\| \sum_{\gamma \in \Lambda} \left( \sum_{(r,c) \in I \atop r-c=\gamma} a_{rc} e_{rc} \right) \gamma \right\|_{L_\Lambda^p(G, S^p(S^p))}. \tag{6}
\end{align}
$$

This yields an isometric embedding of $S_I^p(S^p)$ in $L_\Lambda^p(G, S_I^p(S^p))$. As $S^p(S^p)$ may be identified with $S^p(\ell_\Gamma^2(\ell^2))$,

$$
\left\| \sum_{q \in I} \varphi_q a_q e_q \right\|_{S_I^p(S^p)} = \left\| \sum_{\gamma \in \Lambda} \mu_\gamma \left( \sum_{(r,c) \in I \atop r-c=\gamma} a_{rc} e_{rc} \right) \gamma \right\|_{L_\Lambda^p(G, S^p(S^p))} \le \| \mathrm{Id} \otimes \mathrm{M}_\mu \| \left\| \sum_{q \in I} a_q e_q \right\|_{S_I^p(S^p)}.
$$

(b). Let us embed $L_\Lambda^p(G)$ into $S_I^p$ by $f \mapsto m_f$, where $m_f: \ell_C^2 \to \ell_R^2$ is the convolution operator defined by

$$
m_f e_c = \hat{f} * e_c = \sum_{\gamma \in \Lambda} \hat{f}(\gamma) e_\gamma * e_c = \sum_{r-c \in \Lambda} \hat{f}(r-c) e_r :
$$

$m_f$ 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 $\ell_C^2$ such that $m_f g = f(g)g$: therefore

$$
\|m_f\|_p = \left( \sum_{g \in G} |f(g)|^p \right)^{1/p} = (\#G)^{1/p} \|f\|_{L^p(G)}.
$$

As $M_\varphi m_f = m_{\widehat{M}_\mu f}$, this shows that the norm of $\mu$ on $L_\Lambda^p(G)$ is the norm of $\varphi$ on the subspace of circulant matrices in $S_I^p$. The same holds for complete norms. $\square$
### 3 Idempotent Schur multipliers of norm 1

A Schur multiplier is idempotent if it is the indicator function $\chi_I$ of some set $I \subseteq R \times C$; if $\chi_I$ is a Schur multiplier on $S^p$, then it is a projection of $S^p$ onto $S_I^p$. Idempotent Schur multipliers on $S^p$ and tensors in $\ell_C^\infty \hat{\otimes} \ell_R^\infty$ with 0, 1 coefficients of norm 1 may be characterised by the combinatorics of $I$.

**Proposition 3.1.** Let $I \subseteq R \times C$ be nonempty and $0 < p \neq 2 < \infty$. The following are equivalent.

(a) For every finite rectangle set $R' \times C'$ intersecting $I$

$$ \left\| \sum_{(r,c) \in I \cap R' \times C'} e_c \otimes e_r \right\|_{\ell_C^\infty \hat{\otimes} \ell_R^\infty} = 1. $$

(b) $S_I^p$ is completely 1-complemented in $S^p$.

(c) $S_I^p$ is 1-complemented in $S^p$.

(d) $I$ is a union of pairwise disjoint complete bipartite graphs: there are pairwise disjoint sets $R_j \subseteq R$ and pairwise disjoint sets $C_j \subseteq C$ such that $I = \cup R_j \times C_j$.

Property (d) means that the pattern $I$ is, up to a permutation of columns and rows, block-diagonal:

$$ \begin{array}{c} C_1 \quad C_2 \quad C_3 \quad \dots \\ R_1 \quad \left( \begin{array}{ccccc} * & 0 & 0 & \dots & * \\ 0 & * & 0 & \dots & 0 \\ 0 & 0 & * & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \end{array} \right) \\ R_2 \\ R_3 \\ \vdots \end{array} $$

*Proof.* (b) ⇒ (c) is trivial.

(a) ⇒ (b). The complete norm of a Schur multiplier $\varphi$ on $S^p$ is the supremum of the complete norm of its restrictions $\varphi' = (\varphi_q)_{q \in R' \times C'}$ to finite rectangle sets $R' \times C'$. Furthermore, the complete norm of an elementary Schur multiplier $(\eta_c \zeta_r)_{(r,c) \in R \times C} = \eta \otimes \zeta$ on $S^p$ equals $\|\eta\|_{\ell_C^\infty} \|\zeta\|_{\ell_R^\infty}$.

(c) ⇒ (d). If $I$ is not a union of pairwise disjoint complete bipartite graphs, then there are $r_0, r_1 \in R$ and $c_0, c_1 \in C$ such that

$$ I' = I \cap \{r_0, r_1\} \times \{c_0, c_1\} = \{(r_0, c_0), (r_1, c_0), (r_0, c_1)\}. $$

By Proposition 2.4, the average of a contractive projection of $S^p$ onto $S_I^p$ 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 $x(t), t \in \mathbb{R}$, be the operator from $\ell_C^2$ to $\ell_R^2$ whose matrix coefficients vanish except for its $\{r_0, r_1\} \times \{c_0, c_1\}$ submatrix, which is $\begin{pmatrix} 1 & \sqrt{2} \\ \sqrt{2} & t \end{pmatrix}$. Its eigenvalues are

$$ \frac{1+t+\sqrt{9-2t+t^2}}{2} = 2+\frac{t}{3}+o(t), \quad \frac{1+t-\sqrt{9-2t+t^2}}{2} = -1+\frac{2t}{3}+o(t), $$

so that

$$ \begin{cases} \|x(t)\|_{\infty} = 2 + t/3 + o(t) \\ \|x(t)\|_{p}^{p} = 2^{p} + 1 + p(2^{p} - 4)t/6 + o(t) \quad \text{for } 0 < p < \infty \end{cases} $$

and therefore $\|\chi_{I'} * x(t)\|_p = \|x(0)\|_p > \|x(t)\|_p$ for some $t \neq 0$ if $p \neq 2$.

(d) ⇒ (a). Suppose (d) and let $R' \times C'$ intersect $I$. Then there are pairwise disjoint sets $R'_j$ and pairwise disjoint sets $C'_j$ such that $I \cap R' \times C' = R'_1 \times C'_1 \cup \dots \cup R'_n \times C'_n$ and

$$ \sum_{(r,c) \in I \cap R' \times C'} e_c \otimes e_r = \sum_{j=1}^{n} \chi_{C'_j} \otimes \chi_{R'_j} = \text{average} \left( \sum_{j=1}^{n} \epsilon_j \chi_{C'_j} \right) \otimes \left( \sum_{j=1}^{n} \epsilon_j \chi_{R'_j} \right) $$

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. □
Remark 3.2. Note that the proof of Prop. 3.1 shows that the norm of a projection $M_{\chi_I}: S^\infty \to S_I^\infty$ is either 1 or at least $2/\sqrt{3}$, as

$$ \left\| \begin{pmatrix} 1 & \sqrt{2} \\ \sqrt{2} & -1 \end{pmatrix} \right\|_{\infty} = \sqrt{3}, \qquad \left\| \begin{pmatrix} 1 & \sqrt{2} \\ \sqrt{2} & 0 \end{pmatrix} \right\|_{\infty} = 2. $$

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 $\sqrt{5}/2$ [25, Th. 3.7.2]. The norm of $M_{\chi_I}$ actually equals $2/\sqrt{3}$ for $I = \{(0,0), (0,1), (1,0)\}$, as shown in [13, Lemma 3]. In fact, the following decomposition holds:

$$ e_0 \otimes e_0 + e_0 \otimes e_1 + e_1 \otimes e_0 = \left( \left( (e^{-i\pi/12}, e^{i\pi/4}) \otimes (e^{-i\pi/12}, e^{i\pi/4}) \right) + \left( (e^{i\pi/12}, e^{-i\pi/4}) \otimes (e^{i\pi/12}, e^{-i\pi/4}) \right) \right) / \sqrt{3}. $$

Remark 3.3. Results related to the equivalence of (c) with (d) have been obtained independently by Banks and Harcharras [1].

# 4 Unconditional basic sequences in $S^p$

**Definition 4.1.** Let $0 < p \le \infty$ and $I \subseteq R \times C$. Let $\mathbb{S} = \mathbb{T}$ (vs. $\mathbb{S} = \{-1, 1\}$.)

(a) $I$ is an *unconditional basic sequence* in $S^p$ if there is a constant $D$ such that

$$ \left\| \sum_{q \in I} \epsilon_q a_q e_q \right\|_p \le D \left\| \sum_{q \in I} a_q e_q \right\|_p \quad (7) $$

for every choice of signs $\epsilon_q \in \mathbb{S}$ and every finitely supported sequence of complex coefficients $a_q$. Its *complex (vs. real) unconditional constant* is the least such constant $D$.

(b) $I$ is a completely unconditional basic sequence in $S^p$ if there is a constant $D$ such that (7) holds for every choice of signs $\epsilon_q \in \mathbb{S}$ and every finitely supported sequence of operator coefficients $a_q \in S^p$. Its complex (vs. real) *complete unconditional constant* is the least such constant $D$.

(c) $I$ is a complex (vs. real, complex completely, real completely) 1-unconditional basic sequence in $S^p$ if its complex (vs. real, complex complete, real complete) unconditional constant is 1: Inequality (7) turns into the equality

$$ \left\| \sum_{q \in I} \epsilon_q a_q e_q \right\|_p = \left\| \sum_{q \in I} a_q e_q \right\|_p. $$

If Inequality (7) 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 [28],) so that there is no need to distinguish between complex and real unconditional basic sequences.

The notions defined in (a) and (b) are called $\sigma(p)$ sets and complete $\sigma(p)$ sets in [8, §4] and [9] (see also the survey [23, §9].) The notions defined in (c) are their isometric counterparts.

By [27, proof of Cor. 4], the real unconditional constant of any basis of $S_I^p$ cannot be lower than a fourth of the real unconditional constant of $I$ in $S^p$.

**Example 4.2.** A single column $R \times \{c\}$, a single row $\{r\} \times C$, the diagonal set $\{(\text{row } n, \text{col } n)\}_{n \in \mathbb{N}}$ if $R$ and $C$ are copies of $\mathbb{N}$, are 1-unconditional basic sequences in all $S^p$. In fact, every column section and every row section (this is the terminology of [32, Def. 4.3]) is a 1-unconditional basic sequence; note that the length of every path in the corresponding graph is at most 2.

Note that the set $I$ is a (completely) 1-unconditional basic sequence in $S^p$ if and only if the relative Schur multipliers by signs on $S_I^p$ define (complete) isometries. This yields by Prop. 2.1:

**Proposition 4.3.** Let $I \subseteq R \times C$ and $0 < p \le \infty$. If $I$ is a real (vs. complex) 1-unconditional basic sequence in $S^\infty$, then $I$ is also a real (vs. complex) completely 1-unconditional basic sequence in $S^p$.
*Example 4.4.* If $R = C = \{0, \dots, n-1\}$, $1 \le p \le \infty$ and $I = R \times C$, then the complex unconditional constant of the basis of elementary matrices in $S^p$ is $n^{|1/2-1/p|}$ and coincides with its complete unconditional constant (see [21, Lemma 8.1.5].) 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 $S^p$ is $(2^{|1/2-1/p|})^k = n^{|1/2-1/p|}$. Let us now show that if $n = 3$, the real unconditional constant of the basis of elementary matrices in $S^\infty$ is 5/3 and differs from its complex unconditional constant, $\sqrt{3}$. In fact, because the canonical bases of $\ell_C^2$ and $\ell_R^2$ 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:

$$ \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & -1 \end{pmatrix} \text{ or } \begin{pmatrix} -1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1 \end{pmatrix}. $$

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 $\sqrt{2}$, because the norm of that multiplier equals the norm of its tensor

product by $\mathrm{Id}_{\ell_2^2}$, which is $\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & 1 & -1 & -1 \end{pmatrix}$. By Prop. 2.5 for $\Gamma = \mathbb{Z}/3\mathbb{Z}$, the third one has the

same norm as the Fourier multiplier $\varphi = (-1, 1, 1)$ on $L^\infty(G)$, where $G = \{z \in \mathbb{C} : z^3 = 1\}$: as this multiplier acts by convolution with $f = -1 + z + z^2$, its norm is $\|f\|_{L^1(G)}$, that is

$$ \frac{(|-1+1+1| + |-1+e^{2i\pi/3}+e^{4i\pi/3}| + |-1+e^{4i\pi/3}+e^{2i\pi/3}|)}{3} = 5/3. $$

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 $S^p$ with $p \neq 2$.

# 5 Varopoulos' characterisation of unconditional matrices in $S^\infty$

Our results may be seen as the isometric counterpart to results by Varopoulos [32] on tensor algebras over discrete spaces and their generalisation to $S^p$. He characterised unconditional basic sequences of elementary matrices in $S^\infty$ in his study of the projective tensor product $c_0 \hat{\otimes} c_0$. We gather up his results in the next theorem, as they are difficult to extract from the literature.

**Theorem 5.1.** Let $I \subseteq R \times C$. The following are equivalent.

(a) $I$ is an unconditional basic sequence in $S^\infty$.

(b) $I$ is an interpolation set for Schur multipliers on $S^\infty$: every bounded sequence on $I$ is the restriction of a Schur multiplier on $S^\infty$.

(c) $I$ is a V-Sidon set as defined in [32, Def. 4.1]: every null sequence on $I$ is the restriction of the sequence of coefficients of a tensor in $c_0(C) \hat{\otimes} c_0(R)$.

(d) The coefficients of every tensor in $\ell_C^1 \hat{\otimes} \ell_R^1$ with support in $I$ form an absolutely convergent series.

(e) $(z_c z_r')_{(r,c) \in I}$ is a Sidon set in the dual of $T^C \times T^R$, that is, an unconditional basic sequence in $C(T^C \times T^R)$.

(f) There is a constant $\lambda$ such that for all $R' \subseteq R$ and $C' \subseteq C$ with $n$ elements $\#[I \cap R' \times C'] \le \lambda n$.

(g) $I$ is a finite union of forests.

(h) $I$ is a finite union of row sections and column sections.
(i) Every bounded sequence supported by I is a Schur multiplier on $S^\infty$.

Sketch of proof. (a) ⇒ (b). If (a) holds, every sequence of signs $\epsilon \in \{-1, 1\}^I$ is a Schur multiplier on $S_I^\infty$. By a convexity argument, this implies that every bounded sequence is a Schur multiplier on $S_I^\infty$, which may be extended to a Schur multiplier on $S^\infty$ with the same norm by [17, Cor. 3.3].

(b) ⇒ (c) holds by Grothendieck's inequality (see [22, § 5]) and an approximation argument.

(d) is but the formulation dual to (c) (see [31, § 6.2].)

(d) ⇒ (e). A computation yields

$$ \left\| \sum_{(r,c) \in I} a_{rc} e_c \otimes e_r \right\|_{\ell_C^1 \otimes \ell_R^1} = \sup_{|z_c|, |z'_r| = 1} \left| \sum_{(r,c) \in I} a_{rc} z_c z'_r \right|. \quad (8) $$

(e) ⇒ (f) is [32, Th. 4.2]. (The proof can be found in [31, § 6.3] and in [30, § 5].)

(f) ⇒ (g), (f) ⇒ (h) can be found in [30, Th. 6.1].

(g) ⇒ (h). In fact, a forest is the union of a column section $I'$ with a row section $I''$ (a bisection in the terminology of [32, Def. 4.3].) 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 $(w, w^c)$ with $w$ a word and $c$ a letter is a column section; the set $I''$ of couples of the form $(w^c, w)$ with $w$ a word and $c$ a letter is a row section.

(h) ⇒ (i) is [30, Th. 4.5]. Note that row sections and column sections form 1-unconditional basic sequences in $S^\infty$ and are 1-complemented in $S^\infty$ by Prop. 3.1.

(i) ⇒ (a) follows from the open mapping theorem.

# 6 Closed walk relations

We now introduce and study the combinatorial objects that we need in order to analyse the expansion of the function defined by

$$ \Phi_I(\epsilon, a) = \text{tr} \left| \sum_{q \in I} \epsilon_q a_q e_q \right|^p \quad (9) $$

for $I \subseteq R \times C$, a positive even integer $p = 2k$, signs $\epsilon_q \in \mathbb{T}$ and coefficients $a_q \in \mathbb{C}$, of which only a finite number are nonzero. In fact,

$$ \begin{align} \Phi_I(\epsilon, a) &= \text{tr} \left( \sum_{(r,c), (r',c') \in I} (\epsilon_{rc} a_{rc} e_{rc})^* (\epsilon_{r'c'} a_{r'c'} e_{r'c'}) \right)^k \nonumber \\ &= \text{tr} \sum_{\substack{(r_1, c_1), (r'_1, c'_1), \dots, \\ (r_k, c_k), (r'_k, c'_k) \in I}} \prod_{i=1}^k (\epsilon_{r_i c_i}^{-1} \overline{a_{r_i c_i}} e_{c_i r_i}) (\epsilon_{r'_i c'_i} a_{r'_i c'_i} e_{r'_i c'_i}) \tag{10} \\ &= \sum_{\substack{(r_1, c_1), (r'_1, c'_1), \dots, \\ (r_k, c_k), (r'_k, c'_k) \in I}} \prod_{i=1}^k \epsilon_{r_i c_i}^{-1} \epsilon_{r_i c_{i+1}} \overline{a_{r_i c_i}} a_{r_i c_{i+1}} \quad (\text{where } c_{k+1} = c_1.) \nonumber \end{align} $$

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 $\epsilon^{\beta-\alpha} \bar{a}^{\alpha} a^{\beta}$ with $|\alpha| = |\beta| = k$. The attempt to describe those couples $(\alpha, \beta)$ that effectively arise in this expansion yields the following definition.

**Definition 6.1.** Let $p = 2k \ge 0$ be an even integer and $I \subseteq R \times C$.

$$ \begin{align*} \text{(a) Let } A_k^I &= \{\alpha \in \mathbb{N}^I : \sum_{q \in I} \alpha_q = k\} \text{ and set} \\ B_k^I &= \{ (\alpha, \beta) \in A_k^I \times A_k^I : \forall r \sum_c \alpha_{rc} = \sum_c \beta_{rc} \text{ and } \forall c \sum_r \alpha_{rc} = \sum_r \beta_{rc} \}. \end{align*} $$

$$ \begin{align} \text{(b) Two couples } (\alpha^1, \beta^1) \in B_{k_1}^I, (\alpha^2, \beta^2) \in B_{k_2}^I \text{ are disjoint if } k_1, k_2 \ge 1 \text{ and} \nonumber \\ \alpha_{rc}^1 \ge 1 \Rightarrow \forall (r', c) \in I \quad \alpha_{r'c}^2 = 0 \quad \text{and} \quad \forall (r, c') \in I \quad \alpha_{rc'}^2 = 0. \tag{11} \end{align} $$
(c) The set $W_k^I$ of closed walk relations of length $p$ in $I$ is the subset of those $(\alpha, \beta) \in B_k^I$ that cannot be decomposed into the sum of two disjoint couples.

(d) Let $W_k^I$ 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 $(\alpha, \beta) \in A_k^I \times A_k^I$ defined by

$$ \alpha_q = \#[i \in \{1, \dots, k\} : (r_i, c_i) = q] \\ \beta_q = \#[i \in \{1, \dots, k\} : (r_i, c_{i+1}) = q] \quad (\text{where } c_{k+1} = c_1.) $$

We shall write $P \sim (\alpha, \beta)$ and call $n_{\alpha\beta}$ the number of elements of $W_k^I$ mapped onto $(\alpha, \beta)$.

Note that the conditions in Eq. (11) is in fact symmetric and that it may be stated with $\beta^1$ and $\beta^2$ instead of $\alpha^1$ and $\alpha^2$.

**Example 6.2.** Let $R = C = \{0, 1, 2, 3\}$ and $I = R \times C$. The couple $(e_{00} + e_{11} + e_{22} + e_{33}, e_{01} + e_{10} + e_{23} + e_{32})$ is an element of $B_4^I \setminus W_4^I$: it is the sum of the two disjoint closed walk relations $(e_{00} + e_{11}, e_{01} + e_{10})$ and $(e_{22} + e_{33}, e_{23} + e_{32})$.

**Example 6.3.** Let $I = R \times C = \{0, 1\} \times \{0, 1\}$. Two closed walks are associated with the closed walk relation $(e_{00} + e_{11}, e_{01} + e_{10}) \in W_2^I$: the two cycles $(\text{col } 0, \text{row } 0, \text{col } 1, \text{row } 1)$ and $(\text{col } 1, \text{row } 1, \text{col } 0, \text{row } 0)$. Six closed walks are mapped onto the closed walk relation $(2e_{00} + 2e_{01}, 2e_{00} + 2e_{01})$: the $\frac{4!}{2!2!}$ concatenations of a permutation of $(\text{col } 1, \text{row } 0)$, $(\text{col } 1, \text{row } 0)$, $(\text{col } 0, \text{row } 0)$, $(\text{col } 0, \text{row } 0)$.

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

**Proposition 6.4.** Let $p = 2k \ge 0$ be an even integer and $I \subseteq R \times C$. The image of the mapping in Def. 6.1(d) is $W_k^I$:

(a) if $P \in W_k^I$ and $P \sim (\alpha, \beta)$, then $(\alpha, \beta) \in W_k^I$;

(b) if $(\alpha, \beta) \in W_k^I$, then there is a $P \in W_k^I$ such that $P \sim (\alpha, \beta)$, so that $n_{\alpha\beta} \ge 1$.

*Proof.* (a). Let $P = (c_1, r_1, c_2, r_2, \dots, c_k, r_k)$. In fact,

$$ \sum_c \alpha_{rc} = \#[i \in \{1, \dots, k\} : r_i = r] = \sum_c \beta_{rc} \\ \sum_r \alpha_{rc} = \#[i \in \{1, \dots, k\} : c_i = c] = \sum_r \beta_{rc} $$

and $(\alpha, \beta) \in B_k^I$. If $(\alpha, \beta) = (\alpha^1, \beta^1) + (\alpha^2, \beta^2)$ with $(\alpha^i, \beta^i) \in B_{k_i}^I$ and $k_i \ge 1$, there is an $i$ such that $\alpha_{r_i c_i}^1 \ge 1$ and $\alpha_{r_{i+1} c_{i+1}}^2 \ge 1$ (where $(r_{i+1}, c_{i+1}) = (r_1, c_1)$ if $i = k$.) If $\beta_{r_i c_{i+1}}^1 \ge 1$, then $\sum_r \alpha_{rc_{i+1}}^1 \ge 1$, so that there is an $r$ such that $\alpha_{rc_{i+1}}^1 \ge 1$. Otherwise $\beta_{r_i c_{i+1}}^2 \ge 1$, so that $\sum_c \alpha_{r_i c}^2 \ge 1$ and there is a $c$ such that $\alpha_{r_i c}^2 \ge 1$. Therefore $(\alpha^1, \beta^1)$ and $(\alpha^2, \beta^2)$ are not disjoint and $(\alpha, \beta) \in W_k^I$.

(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 $k \ge 1$; Consider a walk $(c_1, r_1, c_2, r_2, \dots, c_j, r_j, c_{j+1})$ in $I$ such that $\alpha_q^1 = \#[i : (r_i, c_i) = q] \le \alpha_q$ and $\beta_q^1 = \#[i : (r_i, c_{i+1}) = q] \le \beta_q$ for every $q \in R \times C$, 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

$$ \sum_r \alpha_{rc_{j+1}}^1 = \#[i \in \{1, \dots, j\} : c_i = c_{j+1}] \\ \sum_r \beta_{rc_{j+1}}^1 = \#[i \in \{1, \dots, j+1\} : c_i = c_{j+1}] = 1 + \sum_r \alpha_{rc_{j+1}}^1, $$

so that there must be $r_{j+1}$ with $\alpha_{r_{j+1} c_{j+1}}^2 \ge 1$. But then

$$ \sum_c \beta_{r_{j+1}c}^2 = \sum_c \alpha_{r_{j+1}c}^2 \ge 1 $$

and there must be $c_{j+2}$ such that $\beta_{r_{j+1}c_{j+2}}^2 \ge 1$: $j$ is not maximal.
(B). Suppose that $j < k$. Then $(\alpha^1, \beta^1) \in B_j^I$ and $(\alpha^2, \beta^2) \in B_{k-j}^I$. By hypothesis, they are not disjoint: there are $r, c, c'$ such that $\alpha_{rc}^1 \alpha_{rc'}^2 \ge 1$ or $r, r', c$ such that $\alpha_{rc}^1 \alpha_{r',c}^2 \ge 1$. By interchanging $R$ and $C$ 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 $\alpha_{r'_1c'_1}^2 \ge 1$. Then there is $c'_2$ such that $\beta_{r'_1c'_2}^2 \ge 1$. 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 $\#[i : (r'_i, c'_i) = q] \le \alpha_q^2$ and $\#[i : (r'_i, c'_{i+1}) = q] \le \beta_q^2$ (where $c'_{j'+1} = c'_1$.) Then the closed walk

$$ (c_1, r_1, c_2, r_2, \dots, c_j, r_j, c'_2, r'_2, \dots, c'_{j'}, r'_{j'}, c'_1, r'_1) $$

shows that $j$ is not maximal.

<figure><img src="image_4.png" /></figure>

We are now in position to state the following theorem, a matrix counterpart to the computation presented in [14, Prop. 2.5(ii)].

**Theorem 6.5.** Let $p = 2k$ be a positive even integer and $I \subseteq R \times C$.

(a) The function $\Phi_I$ in Eq. (9) has the expansion

$$ \Phi_I(\epsilon, a) = \sum_{(\alpha, \beta) \in \mathcal{W}_k^I} n_{\alpha\beta} \epsilon^{\beta-\alpha} \bar{a}^{\alpha} a^{\beta}, \qquad (12) $$

where $n_{\alpha\beta} \ge 1$ for every $(\alpha, \beta) \in \mathcal{W}_k^I$.

(b) If $\epsilon \in \mathbb{T}^I$ and $a \in (\mathbb{S}^p)^I$ is finitely supported, then the function

$$ \Psi_I(\epsilon, a) = \mathrm{tr} \left| \sum_{q \in I} \epsilon_q a_q e_q \right|^p \qquad (13) $$

has the expansion

$$ \sum_{(\alpha, \beta) \in \mathcal{W}_k^I} \epsilon^{\beta-\alpha} \sum_{(c_1, r_1, \dots, c_k, r_k) \sim (\alpha, \beta)} \prod_{i=1}^k a_{r_i c_i}^* a_{r_i c_{i+1}} \quad (\text{with } c_{k+1} = c_1.) \qquad (14) $$

*Proof.* This follows from Def. 6.1 and Prop. 6.4.

<figure><img src="image_5.png" /></figure>

Note that the edges of a closed walk $P \sim (\alpha, \beta)$ are precisely those $\{r, c\}$ such that $\alpha_{rc} + \beta_{rc} \ge 1$. $P$ is a cycle if and only if $P$ does not have length 0 or 2 and $\sum_r \alpha_{rc} \le 1$ for all $c$ and $\sum_c \alpha_{rc} \le 1$ for all $r$. We now show how to decompose closed walks into cycles.

**Proposition 6.6.** Let $P = (c_1, r_1, c_2, r_2, \dots, c_k, r_k) \sim (\alpha, \beta)$ be a closed walk.

(a) If $r_i = r_j$ (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 $\sum_c \alpha_{rc}^1, \sum_c \alpha_{rc}^2 \ge 1$ (vs. $\sum_r \alpha_{rc}^1, \sum_r \alpha_{rc}^2 \ge 1$.)

(b) $P$ is the juxtaposition of nonempty closed walks $P_j \sim (\alpha^j, \beta^j)$ such that $\sum_r \alpha_{rc}^j \le 1$ for all $c$, $\sum_c \alpha_{rc}^j \le 1$ for all $r$ and $(\alpha, \beta) = \sum (\alpha^j, \beta^j)$.

(c) There are cycles $P_j \sim (\alpha^j, \beta^j)$ and $a \gamma$ such that $(\alpha, \beta) = (\gamma, \gamma) + \sum (\alpha^j, \beta^j)$.

*Proof.* (a). If $r_i = r_j$ for $i < j$, we may suppose that $j = k$: consider the closed walks $P_1 = (c_1, r_1, \dots, c_i, r_i)$ 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 $q \in I$.

<figure><img src="image_6.png" /></figure>
# 7 Schur multipliers on a cycle

We can realise a cycle of even length $2s$, $s \ge 2$, in the following convenient way. Let $\Gamma = \mathbb{Z}/s\mathbb{Z}$. 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 $(\text{col } 0, \text{row } 0, \text{col } 1, \text{row } 1, \dots, \text{col } s-1, \text{row } s-1)$ 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

$$ \begin{array}{c} \begin{array}{ccccccc} 0 & 1 & 2 & \cdots & s-2 & s-1 & \cdots \end{array} \\ \begin{array}{c} 0 \\ 1 \\ 2 \\ \vdots \\ s-2 \\ s-1 \end{array} \begin{pmatrix} * & * & 0 & \cdots & 0 & 0 & \cdots \\ 0 & * & * & \cdots & 0 & 0 & \cdots \\ 0 & 0 & * & \cdots & 0 & 0 & \cdots \\ \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\ 0 & 0 & 0 & \cdots & * & * & \cdots \\ * & 0 & 0 & \cdots & 0 & * & \cdots \end{pmatrix} \end{array} $$

$\Gamma$ is the group dual to $G = \hat{\Gamma} = \{z \in \mathbb{C} : z^s = 1\}$. We shall consider the space $L_{\Lambda}^p(G)$ spanned by $\Lambda = \{1, z\}$ in $L^p(G)$, where $z$ is the identical function on $G$: its norm is given by $\|a + bz\|_{L^p(G)} = (s^{-1} \sum_{z^{s}=1} |a + bz|^p)^{1/p}$.

**Proposition 7.1.** Let $0 < p \le \infty$, $s \ge 2$ and $I = \{(i, i), (i, i+1) : i \in \mathbb{Z}/s\mathbb{Z}\}$. Let $\epsilon \in \mathbb{T}^I$ be a Schur multiplier by signs on $S_I^p$.

(a) 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 $\hat{\epsilon}_{s-1,0} = \overline{\epsilon_{00}\epsilon_{01} \dots \epsilon_{s-1,s-1}\epsilon_{s-1,0}}$.

(b) The Schur multiplier $\epsilon$ has the same norm as $\hat{\epsilon}$ given by $\hat{\epsilon}_{ii} = 1$ and $\hat{\epsilon}_{i,i+1} = \vartheta$ with $\vartheta$ any sth root of $\hat{\epsilon}_{s-1,0}$ or its complex conjugate: without loss of generality, $\vartheta = e^{i\alpha}$ with $\alpha \in [0, \pi/s]$.

(c) The norm of $\epsilon$ on $S_I^p$ is bounded below by the norm of the relative Fourier multiplier $\mu : a + bz \mapsto a + \vartheta bz$ on $L_{\Lambda}^p(G)$; their complete norms are equal.

(d) The norm of $\epsilon$ on $S_I^1$ and on $S_I^{\infty}$ is equal to the norm of $\mu$ on $L_{\Lambda}^1(G)$ and on $L_{\Lambda}^{\infty}(G)$: this norm is

$$ \frac{\cos(\alpha/2 - \pi/2s)}{\cos \pi/2s} = \frac{\max_{z^s=-1} |\vartheta + z|}{|1 + e^{i\pi/s}|} $$

(e) The Schur multiplication operator $M_{\epsilon}$ is an isometry on $S_I^p$ if and only if $p/2 \in \{1, 2, \dots, s-1\}$ or $\overline{\epsilon_{00}\epsilon_{01} \dots \epsilon_{s-1,s-1}\epsilon_{s-1,0}} = 1$.

*Proof.* (a) and (b) follow from the matrix unconditionality of Schatten-von-Neumann norms (see Eq. (4)) and from the fact that the Schur multipliers $\epsilon$ and $\bar{\epsilon} = (\overline{\epsilon_q})_{q \in I}$ have the same norm on $S_I^p$.

(c) follows from Prop. 2.5.

(d). Let us compute $f(\beta) = \|1 + e^{i\beta z}\|_{L^1(G)}$. As $f(\beta) = f(\beta + 2\pi/s) = f(-\beta)$, we may suppose without loss of generality that $\beta \in [0, \pi/s]$. Then $|\beta/2 + k\pi/s| \le \pi/2$ if $-\lfloor s/2 \rfloor \le k \le \lfloor s/2 \rfloor - 1$, so that

$$ \begin{align*} f(\beta) &= \frac{1}{s} \sum_{k=-\lfloor s/2 \rfloor}^{\lfloor s/2 \rfloor-1} |1 + e^{i\beta} e^{2ik\pi/s}| \\ &= \frac{2}{s} \sum_{k=-\lfloor s/2 \rfloor}^{\lfloor s/2 \rfloor-1} \cos(\beta/2 + k\pi/s) \\ &= \frac{2}{s} \Re \left( e^{i\beta/2} \frac{e^{i\lfloor s/2 \rfloor\pi/s} - e^{-i\lfloor s/2 \rfloor/s}}{e^{i\pi/s} - 1} \right) \\ &= \frac{2}{s \sin(\pi/2s)} \cdot \begin{cases} \cos(\beta/2 - \pi/2s) & \text{if } s \text{ is even} \\ \cos(\beta/2) & \text{if } s \text{ is odd.} \end{cases} \end{align*} $$
This shows in both cases that the norm of $\mu$ on $L_{\Lambda}^{1}(G)$ is bounded below by $\cos(\alpha/2 - \pi/2s)/\cos(\pi/2s)$. The complete norm of $\mu$ on $L_{\Lambda}^{\infty}(G)$ is equal to its norm and thus to the maximum of $g(w) = \|w + \vartheta z\|_{L^{\infty}(G)}/\|w + z\|_{L^{\infty}(G)}$ for $w \in \mathbb{C}$. Let $w = re^{i\beta}$ with $r \ge 0$ and $\beta \in \mathbb{R}$. Note that

$$ \|w + z\|_{L^{\infty}(G)} = |r + e^{i\beta(\vartheta, (2\pi/s)\mathbb{Z})}| $$

is a decreasing function of $d(\beta, (2\pi/s)\mathbb{Z})$ and that

$$ d(\alpha - \beta, (2\pi/s)\mathbb{Z}) < d(\beta, (2\pi/s)\mathbb{Z}) \Leftrightarrow \beta \in [\alpha/2, \pi/s + \alpha/2 \bmod 2\pi/s. $$

As $g(w) = g(wz)$ if $z^s = 1$, we may suppose without loss of generality that $\beta \in [\alpha/2, \pi/s + \alpha/2]$. Therefore

$$ g(w) = \begin{cases} \left| \frac{w + e^{i\alpha}}{w + 1} \right| & \text{if } \beta \in [\alpha/2, \pi/s] \\ \left| \frac{w + e^{i\alpha}}{w + e^{2i\pi/s}} \right| & \text{if } \beta \in [\pi/s, \pi/s + \alpha/2]. \end{cases} $$

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,

$$ \begin{aligned} g(re^{i\pi/s})^2 &= \frac{1 + 2r \cos(\pi/s - \alpha) + r^2}{1 + 2r \cos(\pi/s) + r^2} \\ &= 1 + \frac{\cos(\pi/s - \alpha) - \cos \pi/s}{\cos(\pi/s) + (r + 1/r)/2} \le g(e^{i\pi/s})^2 = \left( \frac{\cos(\pi/2s - \alpha/2)}{\cos \pi/2s} \right)^2. \end{aligned} $$

(e). If $p$ is not an even integer and $\vartheta^s \ne 1$, then $\mu$ is not an isometry on $L_{\Lambda}^{p}(G)$: otherwise the functions $z$ and $\vartheta z$ would have the same distribution by the Plotkin-Rudin Equimeasurability Theorem (see [11, Th. 2]). If $p \in \{2, 4, \dots, 2s-2\}$, then $I$ contains no cycle of length 4, 6, ..., $p$, so that by Prop. 6.6(c) every closed walk $P \sim (\alpha, \beta)$ satisfies $\alpha = \beta$. The function $\Phi_I(\epsilon, a)$ in Eq. (9) is therefore constant in $\epsilon$ by Th. 6.5(a). If $p \in \{2s, 2s+2, \dots\}$, the closed walk relation

$$ (\alpha, \beta) = \left( \sum_{i \in \Gamma} e_{ii}, \sum_{i \in \Gamma} e_{i,i+1} \right) + (p/2 - s)(e_{00}, e_{00}) $$

satisfies $n_{\alpha\beta} \ge 1$ by Prop. 6.4. Then the coefficient of $\Phi_I(\epsilon, a)$ in $\bar{a}^{\alpha}a^{\beta}$ equals

$$ n_{\alpha\beta\bar{c}_{00}\epsilon_{01}\dots\bar{c}_{s-1,s-1}\epsilon_{s-1,0}} $$

and must equal the same quantity with $\epsilon$ replaced by 1 if $\epsilon$ defines an isometry on $S_I^p$. $\square$

*Remark 7.2.* See [12, p. 245] for a similar application of the Plotkin-Rudin Equimeasurability Theorem in (e).

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

**Corollary 7.3.** Let $0 < p \le \infty$ and $s \ge 2$. Let $I$ be the cycle of length $2s$.

(a) $I$ is a real 1-unconditional basic sequence in $S^p$ if and only if $p \in \{2, 4, \dots, 2s-2\}$.

(b) The real and complex unconditional constants of $I$ in the spaces $S^1$ and $S^{\infty}$ equal $\sec \pi/2s$.

## 8 1-unconditional matrices in $S^p$, $p$ not an even integer

We now state the announced isometric counterpart to Varopoulos' characterisation of unconditional matrices in $S^{\infty}$ (Section 5) and its generalisation to $S^p$ for $p$ not an even integer.

**Theorem 8.1.** Let $I \subseteq \mathbb{R} \times \mathbb{C}$ be nonempty and $p \in (0, \infty] \setminus 2\mathbb{N}$. The following are equivalent.

(a) $I$ is a complex completely 1-unconditional basic sequence in $S^p$.
(b) I is a complex 1-unconditional basic sequence in $S^p$.

(c) I is a real 1-unconditional basic sequence in $S^p$.

(d) I is a forest.

(e) For each $\epsilon \in \mathbb{T}^I$ there are $\zeta \in \mathbb{T}^C$ and $\eta \in \mathbb{T}^R$ such that $\epsilon_{rc} = \zeta(c)\eta(r)$ for all $(r, c) \in I$.

(f) For each $\epsilon \in \{-1, 1\}^I$ there are $\zeta \in \{-1, 1\}^C$ and $\eta \in \{-1, 1\}^R$ such that $\epsilon_{rc} = \zeta(c)\eta(r)$ for all $(r, c) \in I$.

(g) I is a set of V-interpolation of constant 1: for all $\varphi \in \ell_I^\infty$

$$ \inf \left\{ \left\| \sum_{(r,c) \in R \times C} \tilde{\varphi}_{rc} e_c \otimes e_r \right\|_{\ell_C^\infty \hat{\otimes} \ell_R^\infty} : \tilde{\varphi}|_I = \varphi \right\} = \sup_{q \in I} |\varphi_q|. \quad (15) $$

(h) I is a V-Sidon set of constant 1: for all $\varphi \in c_0(I)$

$$ \inf \left\{ \left\| \sum_{(r,c) \in R \times C} \tilde{\varphi}_{rc} e_c \otimes e_r \right\|_{c_0(C) \hat{\otimes} c_0(R)} : \tilde{\varphi}|_I = \varphi \right\} = \sup_{q \in I} |\varphi_q|. \quad (16) $$

(i) For every tensor $u = \sum_{(r,c) \in I} a_{rc} e_c \otimes e_r$ in $\ell_C^1 \hat{\otimes} \ell_R^1$ with support in $I$ we have $\|u\|_{\ell_C^1 \hat{\otimes} \ell_R^1} = \sum_{(r,c) \in I} |a_{rc}|$.

(j) $(z_c z_r')_{(r,c) \in I}$ is a Sidon set of constant 1 in the dual of $\mathbb{T}^C \times \mathbb{T}^R$, that is, a 1-unconditional basic sequence in $C(\mathbb{T}^C \times \mathbb{T}^R)$: if $(a_{rc})$ is finitely supported,

$$ \sup_{(z,z') \in \mathbb{T}^C \times \mathbb{T}^R} \left| \sum_{(r,c) \in I} a_{rc} z_c z_r' \right| = \sum_{(r,c) \in I} |a_{rc}|. $$

(k) For all $R' \subseteq R$ and $C' \subseteq C$ with $k \ge 1$ elements $\#[I \cap R' \times C'] \le 2k-1$.

(l) I is an isometric interpolation set for Schur multipliers on $S^\infty$: every $\varphi \in \ell_I^\infty$ is the restriction of a Schur multiplier on $S^\infty$ with norm $\|M_\varphi\| = \|\varphi\|_{\ell_I^\infty}$.

*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 $s \ge 2$. Cor. 7.3(a) shows that $I$ is not a real 1-unconditional basic sequence in $S^p$.

(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 $\zeta(c) = \epsilon(r, c)/\eta(r)$. 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 $\eta(r) = \epsilon(r, c)/\zeta(c)$. 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 $R$ and $\zeta$ to $C$ 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 $M_\epsilon$ on any $S_I^p$ is therefore $\|\zeta\|_{\ell_C^\infty} \|\eta\|_{\ell_R^\infty} = 1$.

(e) $\Rightarrow$ (g). If (e) holds, every $\varphi \in \mathbb{T}^I \subseteq \ell_I^\infty$ may be extended to an elementary tensor $\zeta \otimes \eta$ of norm 1. (g) follows because every element of $\ell_I^\infty$ with norm 1 is the half sum of two elements of $\mathbb{T}^I$: note that $e^{it} \cos u = (e^{i(t+u)} + e^{i(t-u)})/2$.

(g) $\Rightarrow$ (h). It suffices to check Equality (16) for $\varphi$ with support contained in a finite rectangle set $R' \times C'$. As $\ell_C^\infty \hat{\otimes} \ell_R^\infty$ is a subspace of $\ell_C^\infty \hat{\otimes} \ell_R^\infty$, Eq. (15) yields Eq. (16).
(h) $\Leftrightarrow$ (i) because they are dual statements.

(i) $\Leftrightarrow$ (j). Use Equality (8).

(h) $\Rightarrow$ (l) may be deduced by the argument of Prop. 3.1(a) $\Rightarrow$ (b).

(l) $\Rightarrow$ (a). Taking sign sequences $\varphi \in \mathbb{T}^I$ in (l) shows that all relative Schur multipliers by signs on $S_I^\infty$ define isometries. Apply Prop. 4.3. $\square$

Remark 8.2. The equivalence of (e) with (j) may also be shown as a consequence of the characterisaiton of Sidon sets of constant 1 in [4].

Let us now answer Question 1.3.

**Corollary 8.3.** Let $I \subseteq \mathbb{R} \times \mathbb{C}$. The following are equivalent.

(a) For all $\varphi \in c_0(I)$ one has $\|\sum_{(r,c) \in I} \varphi_{rc} e_c \otimes e_r\|_{c_0(C) \hat{\otimes} c_0(R)} = \sup_{q \in I} |\varphi_q|.$

(b) There are pairwise disjoint sets $R_j \subseteq \mathbb{R}$ and pairwise disjoint sets $C_j \subseteq \mathbb{C}$ such that $R_j$ or $C_j$ is a singleton for each $j$ and $I = \bigcup R_j \times C_j$: $I$ is the union of the column section $\bigcup_{\# R_j = 1} R_j \times C_j$ with the disjoint row section $\bigcup_{\# R_j > 1} R_j \times C_j$.

(c) $I$ is a union of pairwise disjoint star graphs: every path in $I$ has length at most 2.

*Proof.* (a) $\Rightarrow$ (b) follows from Prop. 3.1(a) $\Rightarrow$ (d) and Th. 8.1(g) $\Rightarrow$ (d).

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

(b) $\Rightarrow$ (a). Suppose (b) and let $\varphi \in c_0(I)$. Let $\alpha_j = \sup_{(r,c) \in R_j \times C_j} |\varphi_{rc}|^{1/2}$. If $\alpha_j = 0$, let us define $\varrho^j = 0$ and $\gamma^j = 0$. Otherwise, if $R_j$ is a singleton $\{r\}$, let us define $\varrho^j = \alpha_j e_r$ and $\gamma^j$ by $\gamma_c^j = \varphi_{rc}/\alpha_j$ if $c \in C_j$ and $\gamma_c^j = 0$ otherwise. Otherwise, $C_j$ is a singleton $\{c\}$ and we define $\gamma^j = \alpha_j e_c$ and $\varrho^j$ by $\varrho_r^j = \varphi_{rc}/\alpha_j$ if $r \in R_j$ and $\varrho_r^j = 0$ otherwise. Note that the $\gamma^j$ have pairwise disjoint support and are null sequences, as well as the $\varrho^j$. Then

$$ \sum_{(r,c) \in I} \varphi_{rc} e_c \otimes e_r = \sum_j \gamma^j \otimes \varrho^j = \text{average} \left( \sum_j \epsilon_j \gamma^j \right) \otimes \left( \sum_j \epsilon_j \varrho^j \right) $$

is an average of elementary tensors in $c_0(C) \hat{\otimes} c_0(R)$ of norm $\sup_{q \in I} |\varphi_q|$, so that this average is also bounded by this norm, which obviously is a lower bound. $\square$

# 9 1-unconditional matrices in $S^p$, p an even integer

Let us now prove Theorem 1.5 as a consequence of Theorem 6.5 together with Proposition 6.6(c).

**Theorem 9.1.** Let $I \subseteq \mathbb{R} \times \mathbb{C}$ and $p = 2k$ a positive even integer. The following assertions are equivalent.

(a) $I$ is a complex completely 1-unconditional basic sequence in $S^p$.

(b) $I$ is a complex 1-unconditional basic sequence in $S^p$.

(c) For every finite subset $F \subseteq I$ there is an operator $x \in S^p$, whose support $S$ contains $F$, such that $\|\sum \epsilon_q x_q e_q\|_p$ does not depend on the complex choice of signs $\epsilon \in \mathbb{T}^S$.

(d) $I$ is a real 1-unconditional basic sequence in $S^p$.

(e) For every finite subset $F \subseteq I$ there is an operator $x \in S^p$ with real matrix coefficients, whose support $S$ contains $F$, such that $\|\sum \epsilon_q x_q e_q\|_p$ does not depend on the real choice of signs $\epsilon \in \{-1, 1\}^S$.

(f) Every closed walk $P \sim (\alpha, \beta)$ of length $2s \le 2k$ in $I$ satisfies $\alpha = \beta$.

(g) $I$ does not contain any cycle of length $2s \le 2k$ as a subgraph.
(h) For each $v, w \in V$ there is at most one path in $I$ of length $l \le k$ that joins $v$ to $w$.

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 $(\alpha, \beta) \in \mathcal{W}_k^S$. Consider $f(\epsilon) = \|\sum \epsilon_q x_q e_q\|_p^p$ as a function on the group $\mathbb{T}^S$. Then the Fourier coefficient $\hat{f}(\epsilon^{\beta-\alpha})$ of $f$ at the Steinhaus character $\epsilon^{\beta-\alpha}$ is, by Th. 6.5(a),

$$ \sum \{n_{\epsilon\zeta} \bar{x}^{\epsilon} x^{\zeta} : (\epsilon, \zeta) \in \mathcal{W}_k^S \text{ and } \zeta - \epsilon = \beta - \alpha\} \\ = \bar{x}^{\gamma} x^{\delta} \sum \{n_{\epsilon\zeta} \bar{x}^{\epsilon-\gamma} x^{\zeta-\delta} : (\epsilon, \zeta) \in \mathcal{W}_k^S \text{ and } \zeta - \delta = \epsilon - \gamma\}. $$

(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 $f(\epsilon) = \|\sum \epsilon_q x_q e_q\|_p^p$ as a function on the group $\{-1, 1\}^S$. Then the Fourier coefficient $\hat{f}(\epsilon^{\beta-\alpha})$ of $f$ at the Walsh character $\epsilon^{\beta-\alpha}$ is, by Th. 6.5(a),

$$ \sum \{n_{\epsilon\zeta} x^{\epsilon+\zeta} : (\epsilon, \zeta) \in \mathcal{W}_k^S \text{ and } \zeta - \epsilon \equiv \beta - \alpha \pmod{2}\} \\ = x^{\gamma+\delta} \sum \{n_{\epsilon\zeta} x^{\epsilon+\zeta-\gamma-\delta} : (\epsilon, \zeta) \in \mathcal{W}_k^S \text{ and } \zeta - \epsilon \equiv \delta - \gamma \pmod{2}\}. $$

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. 6.6(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 $2s \le 2k$.

(f) $\Rightarrow$ (a) holds by Theorem 6.5(b): If each $(\alpha, \beta) \in \mathcal{W}_k^I$ satisfies $\alpha = \beta$, then Eq. (14) shows that $\Psi_I(\epsilon, z)$ as defined in Eq. (13) is constant in $\epsilon$. $\square$

Remark 9.2. The equivalence (b) $\Leftrightarrow$ (g) is a noncommutative analogue to [14, Prop. 2.5(ii)].

Remark 9.3. In [15, Th. 2.7], the condition of Th. 9.1(f) is visualised in another way: a closed walk $P = (c_1, r_1, \dots, c_s, r_s) \sim (\alpha, \beta)$ in $\mathbb{N} \times \mathbb{N}$ is considered as the polygonal closed curve $\gamma$ in $\mathbb{C}$ with sides parallel to the coordinate axes whose successive vertices are $r_1 + ic_1, r_1 + ic_2, r_2 + ic_2, \dots, r_{s-1} + ic_s, r_s + ic_s, r_s + ic_1$ and again $r_1 + ic_1$. 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.

Remark 9.4. One cannot drop the assumption that $x$ has real matrix coefficients in Th. 9.1(e). Consider a $2 \times 2$ matrix $x$. Then $\text{tr} x^* x = \sum |x_q|^2$ 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. $x = \begin{pmatrix} 1 & 1 \\ 1 & i \end{pmatrix}$, 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.

Remark 9.5. Theorem 9.1(h) $\Rightarrow$ (a) is the isometric counterpart to [9, Th. 3.1], which shows in particular that $I$ is an unconditional basic sequence in $\mathbb{S}^{2k}$ 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 [2] have deduced from [24] that it is not so.
## 10 Metric unconditional approximation property for $S_I^p$

Let $R, C$ be two copies of $\mathbb{N}$. It is well known that, apart from $S^2$, no $S^p$ has an unconditional basis or just a local unconditional structure (see [23, § 4].) $S^1$ and $S^\infty$ cannot even be embedded in a space with unconditional basis. If $1 < p < \infty$, then $S^p$ has the unconditional finite dimensional decomposition

$$ \bigoplus_{n \in \mathbb{N}} S_{\{(r,c): r \le n, c=n\}}^p \oplus S_{\{(r,c): r=n+1, c \le n\}}^p $$

because the triangular projection associated to the idempotent Schur multiplier $(\chi_{r \le c})$ is bounded on $S^p$.

**Definition 10.1.** Let $X$ be a separable Banach space and $\mathcal{S} = \mathbb{T}$ (vs. $\mathcal{S} = \{-1, 1\}$.)

* A sequence $(T_k)$ of operators on $X$ is an *approximating sequence* if each $T_k$ has finite rank and $\|T_k x - x\| \to 0$ for every $x \in X$. An approximating sequence of commuting projections is a *finite-dimensional decomposition*.

* ([18].) The *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 $k \ge 2$. $X$ has the *unconditional approximation property (uap)* if there is an approximating sequence $(T_k)$ such that for some constant $D$

$$ \left\| \sum_{k=1}^{n} \epsilon_k \Delta T_k \right\| \le D \quad \text{for all } n \text{ and } \epsilon_k \in \mathcal{S}. $$

The *complex* (vs. *real*) *unconditional constant* of $(T_k)$ is the least such constant $D$.

* ([5, § 3], [7, § 8].) $X$ has the *complex* (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 [5, Th. 3.8] and [7, Lemma 8.1], this is the case if and only if there is an approximating sequence $(T_k)$ such that

$$ \sup_{\epsilon \in \mathcal{S}} \|T_k + \epsilon(\mathrm{Id} - T_k)\| \to 1. \qquad (17) $$

$X$ 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 [6, Cor. IV.4]. 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.

**Example 10.2.** The simplest example is the subspace in $S^p$ of operators with an upper triangular matrix. In fact, if $I \subseteq R \times C$ is such that all columns $I \cap R \times \{c\}$ (vs. all rows $I \cap \{r\} \times C$) are finite, then $S_I^p$ admits a 1-unconditional finite-dimensional decomposition in the corresponding finitely supported idempotent Schur multipliers $\chi_{I \cap R \times \{c\}}$ (vs. $\chi_{I \cap \{r\} \times C}$.)

Our results on complete 1-unconditional basic sequences yield the following theorem.

**Theorem 10.3.** Let $1 \le p \le \infty$. Let $R_r \subseteq R$, $r \in \mathbb{N}$, be pairwise disjoint and finite. Let $C_c \subseteq C$, $c \in \mathbb{N}$, be pairwise disjoint and finite. Let $J \subseteq \mathbb{N} \times \mathbb{N}$ and $I = \bigcup_{(r,c) \in J} R_r \times C_c$. Then the sequence of Schur multipliers $(\chi_{R_r \times C_c})_{(r,c) \in J}$ forms a complex 1-unconditional finite-dimensional decomposition for $S_I^p$ if and only if $J$ is a forest or $p$ is an even integer and $J$ contains no cycle of length 4, 6, ..., $p$.

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

**Proposition 10.4.** Let $1 \le p \le \infty$ and $I \subseteq R \times C$. Let $(T_n)$ be an approximating sequence on $S_I^p$. Then there is a sequence of Schur multipliers $(\varphi_n)$ such that $(M_{\varphi_n})$ is an approximating sequence on $S_I^p$ and such that if $(T_n)$ satisfies (17), then so does $(M_{\varphi_n})$.
Proof. Let $\delta_n > 0$ be such that $\delta_n \to 0$. As $T_n$ has finite rank, there is a finite $R_n \times C_n \subseteq R \times C$ such that the projection $P_{R_n \times C_n}$ of $S^p$ onto $S_{R_n \times C_n}^p$ defined by the Schur multiplier $\chi_{C_n} \otimes \chi_{R_n}$ satisfies $\|P_{R_n \times C_n}T_n - T_n\| < \delta_n$. Let $\varphi_n$ be the Schur multiplier associated to $[T_n]_{R_n \times C_n}$. With the notation of Eq. (5),

$$M_{\varphi_n}(x) - x = \int_{\mathbb{T}^R} d\eta \int_{\mathbb{T}^C} d\zeta M_{\zeta^* \otimes \eta^*}(P_{R_n \times C_n} T_n - \mathrm{Id})(M_{\zeta \otimes \eta} x).$$

As $P_{R_n \times C_n} T_n$ tends to the identity uniformly on compact sets, this shows that $M_{\varphi_n}$ is an approximating sequence. As

$$M_{\varphi_n} + \epsilon(\mathrm{Id} - M_{\varphi_n}) = [P_{R_n \times C_n} T_n + \epsilon(\mathrm{Id} - P_{R_n \times C_n} T_n)],$$

the norm of this operator is at most $\|T_n + \epsilon(\mathrm{Id} - T_n)\| + 2\delta_n$.

<figure><img src="image_7.png" /></figure>

This proposition shows together with Prop. 2.1 the following results.

**Corollary 10.5.** Let $1 \le p \le \infty$ and $I \subseteq R \times C$.

* If $S_I^p$ has (muap), then some sequence of Schur multipliers realises it.

* Let $J \subseteq I$. If $S_I^p$ has (muap), then so does $S_J^p$.

* If $S_I^\infty$ has (muap), then so does $S_I^p$.

Let us define the following asymptotic properties.

**Definition 10.6.** Let $1 \le p \le \infty$, $I \subseteq R \times C$ and $S = \mathbb{T}$ (vs. $S = \{-1, 1\}$.)

* $S_I^p$ is asymptotically unconditional if for every $x \in S_I^p$ and for every bounded sequence $(y_n)$ in $S_I^p$ such that each matrix coefficient of $y_n$ tends to 0

$$\max_{\epsilon \in S} \|x + \epsilon y_n\|_p - \min_{\epsilon \in S} \|x + \epsilon y_n\|_p \to 0.$$

* $I$ enjoys the property ($\mathcal{W}$) of block unconditionality in $S^p$ if for each $\delta > 0$ and finite $F \subseteq I$, there is a finite $G \subseteq I$ such that

$$\forall x \in B_{S_F^p} \quad \forall y \in B_{S_{I \setminus G}^p} \quad \max_{\epsilon \in S} \|x + \epsilon y\|_p - \min_{\epsilon \in S} \|x + \epsilon y\|_p < \delta.$$

The arguments of [14, § 6.2] show mutatis mutandis

**Theorem 10.7.** Let $1 \le p \le \infty$, $I \subseteq R \times C$ and $S = \mathbb{T}$ (vs. $S = \{-1, 1\}$.) Consider the following properties.

(a) $S_I^p$ is asymptotically unconditional.

(b) $I$ enjoys ($\mathcal{W}$) in $S^p$.

(c) $S_I^p$ has (muap).

Then (c) $\Rightarrow$ (a) $\Leftrightarrow$ (b). If $1 < p < \infty$, then (b) $\Leftrightarrow$ (c). If $p = 1$, $S_I^1$ has (muap) if and only if $S_I^1$ has (uap) and $I$ enjoys ($\mathcal{W}$) in $S^1$.

The case $p = \infty$ is extreme in the sense that the following properties are equivalent for $S_I^\infty$: to be a dual space, to be reflexive, to have a finite cotype, not to contain $c_0$, 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 $S_I^\infty$ is isomorphic to $\ell_I^2$.

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

**Definition 10.8.** Let $I \subseteq R \times C$ and $k \ge 1$.
• $I$ enjoys property $\mathcal{J}_k$ 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 $R' \times C'$ such that $P$ cannot be completed with edges in $I \setminus R' \times C'$ to a cycle of length $2s \in \{4j + 2, \dots, 2k\}$.

• The asymptotic distance $d_\infty(r, c)$ of $r \in R$ and $c \in C$ in $I$ is the supremum, over all finite rectangle sets $R' \times C'$, of the distance from $r$ to $c$ in $I \setminus R' \times C'$.

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

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

**Proposition 10.9.** Let $I \subseteq R \times C$ and $k \ge 1$.

(a) $I$ enjoys $\mathcal{J}_k$ if and only if any two vertices $r \in R$ and $c \in C$ at distance $2j+1 \le k$ satisfy $d_\infty(r, c) \ge 2k - 2j + 1$.

(b) If $d_\infty(r, c) \ge 2k+1$ for all $(r, c) \in R \times C$, then $I$ enjoys $\mathcal{J}_k$.

(c) If $d_\infty(r, c) \le k$ for some $(r, c) \in R \times C$, then $I$ fails $\mathcal{J}_k$.

(d) $I$ enjoys $\mathcal{J}_k$ for every $k$ if and only if $d_\infty(r, c) = \infty$ for every $(r, c) \in R \times C$.

*Proof.* (a) is but a reformulation of the definition of $\mathcal{J}_k$ 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 $(c, r_1, c_1, \dots, r_j, c_j, r)$ that can be completed with edges outside any given finite set to a cycle of length $4j+2 \le 2k$. $\square$

**Theorem 10.10.** Let $I \subseteq R \times C$ and $1 \le p \le \infty$. If $p$ is an even integer, then $S_I^p$ has complex or real (muap) if and only if $I$ enjoys $\mathcal{J}_{p/2}$. If $p = \infty$ or if $p$ is not an even integer, then $S_I^p$ has real (muap) only if $I$ enjoys $\mathcal{J}_k$ for every $k$.

*Proof.* Suppose that $I$ enjoys $(\mathcal{W})$ in $S^p$ and fails $\mathcal{J}_k$. Then, for some $s \le k$, $I$ contains a sequence of cycles $(c_0, r_0, \dots, c_{j-1}, r_{j-1}, c_j^n, r_j^n, \dots, c_{s-1}^n, r_{s-1}^n)$ with the property that $\|x-y\|_p \le (1+1/n)\|x+y\|_p$ for all $x$ with support in $\{(r_0, c_0), (r_0, c_1), \dots, (r_{j-2}, c_{j-1}), (r_{j-1}, c_{j-1})\}$ and all $y$ with support in $\{(r_{j-1}, c_j^n), (r_j^n, c_j^n), \dots, (r_{s-1}^n, c_{s-1}^n), (r_{s-1}^n, c_0)\}$. With the notation of Section 7, this amounts to stating that the multiplier on $I = \{(i, i), (i, i+1)\} \subseteq \mathbb{Z}/s\mathbb{Z} \times \mathbb{Z}/s\mathbb{Z}$ given by $\epsilon_{rc} = 1$ if $r, c \in \{0, \dots, j-1\}$ and $\epsilon_{rc} = -1$ otherwise actually is an isometry on $S_I^p$. As $\epsilon_{00}\epsilon_{01}\dots\epsilon_{s-1s-1}\epsilon_{s-10} = (-1)^{2s-2j+1} = -1$, this implies by Prop. 7.1(e) that $p/2 \in \{1, 2, \dots, s-1\}$.

Suppose that $I$ enjoys $\mathcal{J}_k$. 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 $v_{b_i}, \dots, v_{a_{i+1}-1} \in F$,

$$ \{i \in \{0, \dots, m\} : a_i, b_i \text{ even}\} = \{i \in \{0, \dots, m\} : a_i, b_i \text{ odd}\}. $$

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 $(v_{a_0}^n, \dots, v_{b_0-1}^n, v_{b_0}, \dots, v_{a_{1}-1}, \dots, v_{a_m}^n, \dots, v_{b_m-1}^n, v_{b_m}, \dots, v_{2s-1})$ such that the $(v_i^n)_{n \ge 0}$ 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 $P = (v_{b_0}, \dots, v_{a_0-1}, v_{a_0}^0, \dots, v_{b_m-1}^0, v_{b_m}, \dots, v_{2s-1})$ of odd length $2s-1 - (b_0 - a_0)$; if $b_0 - a_0 \le s-1$, consider the path $P = (v_{2s-1}, v_{a_0}^0, \dots, v_{b_0-1}^0, v_{b_0})$ 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 $(v_{2s-1}, v_{a_0}^n, \dots, v_{b_0-1}^n, v_{b_0})$ is a path of length $b_0 - a_0 + 1$ in $I$ for every $n$. This proves that $I$ fails $\mathcal{J}_s$.
The claim shows that $I$ enjoys $(\mathcal{W})$ in $S^p$ for $p = 2k$. In fact, if $\tilde{\epsilon} \in \mathbb{T}^{F \cup (I \setminus G)}$ is defined by $\tilde{\epsilon}_q = 1$ for $q \in F$ and $\tilde{\epsilon}_q = \epsilon \in \mathbb{T}$ for $q \in I \setminus G$, then, with the notation of Th. 6.5,

$$ \Phi_{F \cup (I \setminus G)}(\tilde{\epsilon}, a) = \sum_{(\alpha, \beta) \in \mathcal{W}_k^{F \cup (I \setminus G)}} n_{\alpha\beta} \epsilon \sum_{q \in I \setminus G} \beta_q - \alpha_q \bar{a}^\alpha a^\beta $$

does not depend on $\epsilon$, so that $\|x + \epsilon y\|_{2k} = \|x + y\|_{2k}$ if $x \in S_F^{2k}$ and $y \in S_{I \setminus G}^{2k}$, and $S_I^{2k}$ has complex (muap) by Th. 10.7(b) $\Rightarrow$ (c). $\square$

*Remark 10.11.* This theorem is a noncommutative analogue to [14, Th. 7.5].

# 11 Examples

One of Varopoulos' motivations for the study of the projective tensor product $\ell_\infty \hat{\otimes} \ell_\infty$ 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 *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 $\{j^k\}_{k \ge 0}$ with $j \in \{2, 3, \dots\}$ is *n-independent* in $\mathbb{Z}$ if and only if $j \ge n$ [14, § 3]. If $\Lambda$ is *n-independent* for all $n$, then $\Lambda$ is *independent*. Let

$$ Z_n = \{ \zeta \in \mathbb{Z}^\Lambda : \sum_{\gamma \in \Lambda} \zeta_\gamma = 0 \text{ and } \sum_{\gamma \in \Lambda} |\zeta_\gamma| \le 2n \} $$

and $Z = \bigcup Z_n$. Then $\Lambda$ is *n-independent* if and only if, for every $\zeta \in Z_n$,

$$ \sum_{\gamma \in \Lambda} \zeta_{\gamma} \gamma = 0 \Rightarrow \zeta = 0 $$

and $\Lambda$ is *independent* if and only if this holds for every $\zeta \in Z$.

Let us say that $\Lambda$ is *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 $\zeta \in Z_n$,

$$ \sum_{\gamma \in \Lambda} \zeta_{\gamma} \gamma = 0 \implies \forall \gamma \in \Lambda \quad \zeta_{\gamma} = 0 \pmod{2}; $$

$\Lambda$ is *independent modulo 2* if this holds for every $\zeta \in Z$. 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 [14, Prop. 2.5(ii)] for the case $\Gamma = \mathbb{Z}$ shows that $\Lambda$ is a complex (vs. real) 1-unconditional basic sequence in $L^p(G)$ with $p \in 2\mathbb{N}^*$ if and only if $\Lambda$ is $p/2$-independent (vs. modulo 2). Furthermore $\Lambda$ is a complex (vs. real) 1-unconditional basic sequence in $L^p(G)$ with $p \in (0, \infty] \setminus 2\mathbb{N}^*$ if and only if $\Lambda$ is independent (vs. modulo 2). If $\Gamma$ contains no element of order 2, then a real 1-unconditional basic sequence in $L^p(G)$ is also complex 1-unconditional. All these results hold also for the complete counterparts to 1-unconditional basic sequences.

Results on lacunary sets in a discrete abelian group transfer to lacunary matrices in the following way, as in [30, Th. 4.2].

**Proposition 11.1.** Let $\Gamma$ be a discrete abelian group and $R, C$ be countable subsets of $\Gamma$. To every $\Lambda \subseteq R + C$ associate $I_\Lambda = \{(r, c) \in R \times C : r + c \in \Lambda\}$. Let $G = \hat{\Gamma}$.

(a) If $\Lambda$ is a complex 1-unconditional basic sequence in $L^4(G)$, then $I_\Lambda$ is a 1-unconditional basic sequence in $S^4$.

(b) Suppose that each element of $\Gamma$ admits at most one representation as the sum of an element of $R$ with an element of $C$. Then every $I \subseteq R \times C$ has the form $I = I_\Lambda$ with $\Lambda = \{r + c : (r, c) \in I\}$. If $\Lambda$ is a real 1-unconditional basic sequence in $L^p(G)$, then $I_\Lambda$ is a 1-unconditional basic sequence in $S^p$.
(c) Let $p = 2k$ be a positive even integer. Suppose that $R \cap C = \emptyset$ and $R \cup C$ is $k$-independent modulo 2. $I_\Lambda$ is a 1-unconditional basic sequence in $S^p$ if and only if $\Lambda$ is a real 1-unconditional basic sequence in $L^p(G)$.

*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-unconditional basic sequence in $L^p(G)$, then it is also a complete real 1-unconditional basic sequence in $L^p(G)$. Let $\varphi \in \{-1, 1\}^{I_\Lambda}$, so that $\varphi_{q_\gamma} \in \{-1, 1\}$ for all $\gamma \in \Lambda$. Then, as in Eq. (6),

$$ \left\| \sum_{q \in I_\Lambda} a_q e_q \right\|_{S_{I_\Lambda}^p(S^p)} = \left\| \sum_{(r,c) \in I_\Lambda} r(g)c(g)a_{rc}e_{rc} \right\|_{S_{I_\Lambda}^p(S^p)} \\ = \left\| \sum_{\gamma \in \Lambda} a_{q_\gamma} e_{q_\gamma} \gamma(g) \right\|_{S_{I_\Lambda}^p(S^p)} = \left\| \sum_{\gamma \in \Lambda} a_{q_\gamma} e_{q_\gamma} \gamma \right\|_{L_{\Lambda}^p(G, S^p(S^p))} $$

so that as in Eq. (2), by complete real 1-unconditionality of $\Lambda$ in $L^p(G)$,

$$ \left\| \sum_{q \in I_\Lambda} \varphi_q a_q e_q \right\|_{S_{I_\Lambda}^p(S^p)} = \left\| \sum_{\gamma \in \Lambda} \varphi_{q_\gamma} a_{q_\gamma} e_{q_\gamma} \gamma \right\|_{L_{\Lambda}^p(G, S^p(S^p))} = \left\| \sum_{q \in I_\Lambda} a_q e_q \right\|_{S_{I_\Lambda}^p(S^p)} $$

(c). Each element of $\Gamma$ admits at most one representation as the sum of an element of $R$ with an element of $C$, so that (b) yields sufficiency. Suppose that $\Lambda$ is not a real 1-unconditional basic sequence in $L^p(G)$ and let $\zeta \in Z_k$ such that $\sum_{\gamma \in \Lambda} \zeta_\gamma \gamma = 0$ and $J = \{(r, c) \in I_\Lambda : \zeta_{r+c} \neq 0 \text{ (mod 2)}\}$ 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 $R \cup C$ 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 $R \cap C = \emptyset$, $v_{j+1} = v_{j+1-2i}$ 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-unconditional basic sequence in $S^p$. $\square$

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

**Corollary 11.2.** Let $I \subseteq R \times C$ and $p \in (0, \infty]$. The following are equivalent:

* $I$ is a 1-unconditional basic sequence in $S^p$.

* $\{\epsilon_c \epsilon'_r : (r, c) \in I\}$ is a real 1-unconditional basic sequence in $L^p(G)$.

* $\{z_c z'_r : (r, c) \in I\}$ is a 1-unconditional basic sequence in $L^p(\mathbb{T}^C \times \mathbb{T}^R)$.

**Remark 11.3.** The isomorphic counterpart is also true: $I$ is a completely unconditional basic sequence in $S^p$ (i.e., a complete $\sigma(p)$ set) if and only if $\{\epsilon_c \epsilon'_r : (r, c) \in I\}$ is a completely unconditional basic sequence in $L^p(G)$ (a $\Lambda(p)_{cb}$ set in $\hat{G}$, see [8] and [21, §8.1],) if and only if $\{z_c z'_r : (r, c) \in I\}$ is a completely unconditional basic sequence in $L^p(\mathbb{T}^C \times \mathbb{T}^R)$. This follows e.g. from the proof of Prop. 11.1(b) and the iterated noncommutative Khinchin inequality [21, Eq. (8.4.11)].

Harcharras [8] used Peller's discovery [19] of the link between Fourier and Hankel Schur multipliers to produce unconditional basic sequences in $S^p$ that are unions of antidiagonals in $\mathbb{N} \times \mathbb{N}$. We have in our context the rather disappointing

**Proposition 11.4.** Let $\Lambda \subseteq \mathbb{N} \subseteq \mathbb{Z}$ and $I = \{(r, c) \in \mathbb{N} \times \mathbb{N} : r+c \in \Lambda\}$.

(a) $I$ is a 1-unconditional basic sequence in $S^4$ if and only if $\{z^\lambda\}_{\lambda \in \Lambda}$ is a 1-unconditional basic sequence in $L^4(\mathbb{T})$.
(b) If $\Lambda$ contains three elements $\lambda < \mu < \nu$ such that $\lambda + \mu \ge \nu$, then $I$ is not a 1-unconditional basic sequence in $S^p$ if $p \in (0, \infty] \setminus \{2, 4\}$.

(c) If $\Lambda = \{\lambda_k\}$ with $\lambda_{k+1} > 2\lambda_k$ for all $k$, then $I$ is a 1-unconditional basic sequence in $S^p$ for every $p$.

*Proof.* (a). Sufficiency follows from Prop. 11.1(a) with $R = C = \mathbb{N}$. Conversely, if $\Lambda$ contains a solution to $\lambda + \mu = \lambda' + \mu'$ with $\lambda < \lambda' \le \mu' < \mu$, then $I$ contains the cycle (col 0, row $\lambda$, col $\lambda' - \lambda$, row $\mu'$).

(b). Consider the cycle (col 0, row $\lambda$, col $\nu - \lambda$, row $\mu - \nu + \lambda$, col $\nu - \mu$, 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 $c_1 + r_2 < 0$. $\square$

*Remark 11.5.* Further computations yield the following result. If $\{z^\lambda\}_{\lambda \in \Lambda}$ is a 1-unconditional basic sequence in $L^6(\mathbb{T})$ and if $\{\lambda < \mu < \nu\} \subseteq \Lambda \Rightarrow \lambda + \mu < \nu$, then $I$ is a 1-unconditional basic sequence in $S^6$; the converse does not hold.

Let us now give an overview of the known extremal bipartite graphs without cycle of length 4, 6, ..., 2k and their size. Look up [3, Def. I.3.1] for the definition of a Steiner system and [29, Def. 1.3.1] for the definition of a generalised polygon. An elementary example is given in the introduction with (1).

**Proposition 11.6.** Let $2 \le n \le m$, $I \subseteq R \times C$ with $\#C = n$ and $\#R = m$, and $e = \#I$.

(a) If $I$ is a 1-unconditional basic sequence in $S^4$, then

$$n \ge 1 + \left(\frac{e}{m} - 1\right) + \left(\frac{e}{m} - 1\right)\left(\frac{e}{n} - 1\right),$$

that is $e^2 - me - mn(n-1) \le 0$. Equality holds if and only if $I$ is the incidence graph of a Steiner system $S(2, e/m; n)$ on $n$ points and $m$ blocks.

(b) If $I$ is a 1-unconditional basic sequence in $S^6$, then

$$n \ge 1 + \left(\frac{e}{m} - 1\right) + \left(\frac{e}{m} - 1\right)\left(\frac{e}{n} - 1\right) + \left(\frac{e}{m} - 1\right)^2\left(\frac{e}{n} - 1\right),$$

that is $e^3 - (m+n)e^2 + 2mne - m^2n^2 \le 0$. Equality holds if and only if $I$ is the incidence graph of the quadrangle (the cycle of length 8) or of a generalised quadrangle with $n$ points and $m$ lines.

(c) If $I$ is a 1-unconditional basic sequence in $S^{2k}$ with $k \ge 1$ an integer, then

$$n \ge \sum_{i=0}^{k} \left(\frac{e}{m} - 1\right)^{\lfloor \frac{i}{2} \rfloor} \left(\frac{e}{n} - 1\right)^{\lfloor \frac{i}{2} \rfloor}. \qquad (18)$$

Equality holds if $I$ is the incidence graph of the $(k+1)$-gon (the cycle of length $2k+2$) or of a generalised $(k+1)$-gon with $n$ points and $m$ lines.

*Proof.* By Theorem 9.1(b) ⇒ (g), $I$ is a 1-unconditional basic sequence in $S^{2k}$, with $k \ge 1$ an integer, if and only if $I$ is a graph of girth $2k+2$ in the sense of [10]. Therefore (a) and (b) are shown in [16, Prop. 4, Th. 8, Rem. 10]. Inequality (18) is [10, Eq. (1)] and the sufficient condition for equality follows from [29, Lemma 1.5.4]. $\square$
Consult [3, Tables A1.1, A5.1] for examples of Steiner systems and [29, Table 2.1] for examples of generalised polygons. In both cases, the corresponding incidence graph is biregular: every vertex in $R$ has same degree $s+1$ and every vertex in $C$ has same degree $t+1$. Arbitrarily large generalised $(k+1)$-gons exist only if $2k \in \{4, 6, 10, 14\}$ [29, Lemma 1.7.1]; for $2k \in \{6, 10, 14\}$, it follows from [29, Lemma 1.5.4] that

$$n = (s+1) \frac{(st)^{(k+1)/2} - 1}{st - 1}, \quad m = (t+1) \frac{(st)^{(k+1)/2} - 1}{st - 1}.$$

*Remark 11.7.* Let $I \subseteq R \times C$ with $\#C = \#R = n$. Inequality (18) shows that if $I$ is a 1-unconditional basic sequence in $S^{2k}$, then $\#I \le n^{1+1/k} + (s-1)n/s$. If $p \notin \{4, 6, 10\}$, the existence of 1-unconditional basic sequences in $S^{2k}$ such that $\#I \ge n^{1+1/k}$ is in fact an important open problem in graph theory: extremal graphs cannot correspond to generalised polygons and necessarily have less structure.

**Acknowledgement.** I would like to thank Éric Ricard for his comments and suggestions.

## Bibliography

[1] William D. Banks and Asma Harcharras. On the norm of an idempotent Schur multiplier on the Schatten class. Proc. Amer. Math. Soc., 132(7):2121-2125, 2004. (p. 9).

[2] William D. Banks, Asma Harcharras, Ilijas Farah, and Dominique Lecomte. The R-C problem. manuscript, 2003. (p. 18).

[3] T. Beth, D. Jungnickel, and H. Lenz. Design theory. Cambridge University Press, second edition, 1999. (pp. 24 and 25).

[4] Donald I. Cartwright, Robert B. Howlett, and John R. McMullen. Extreme values for the Sidon constant. Proc. Amer. Math. Soc., 81(4):531-537, 1981. (p. 17).

[5] P. G. Casazza and N. J. Kalton. Notes on approximation properties in separable Banach spaces. In P. F. X. Müller and W. Schachermayer, editors, Geometry of Banach spaces (Strobl, 1989), London Math. Soc. Lect. Notes 158, pages 49-63. Cambridge Univ. Press, 1991. (p. 19).

[6] G. Godefroy and N. J. Kalton. Approximating sequences and bidual projections. Quart. J. Math. Oxford (2), 48:179-202, 1997. (p. 19).

[7] G. Godefroy, N. J. Kalton, and P. D. Saphar. Unconditional ideals in Banach spaces. Studia Math., 104:13-59, 1993. (p. 19).

[8] Asma Harcharras. Fourier analysis, Schur multipliers on $S^p$ and non-commutative $\Lambda(p)$-sets. Studia Math., 137(3):203-260, 1999. (pp. 9 and 23).

[9] Asma Harcharras, Stefan Neuwirth, and Krzysztof Oleszkiewicz. Lacunary matrices. Indiana Univ. Math. J., 50:1675-1689, 2001. (pp. 9 and 18).

[10] Shlomo Hoory. The size of bipartite graphs with a given girth. J. Combin. Theory Ser. B, 86(2):215-220, 2002. (p. 24).

[11] Alexander Koldobsky and Hermann König. Aspects of the isometric theory of Banach spaces. In W. B. Johnson and J. Lindenstrauss, editors, Handbook of the geometry of Banach spaces, Vol. 1, pages 899-939. North-Holland, 2001. (p. 15).

[12] Daniel Li. Complex unconditional metric approximation property for $C_\Lambda(\mathbb{T})$ spaces. Studia Math., 121(3):231-247, 1996. (p. 15).

[13] Leo Livshits. A note on 0-1 Schur multipliers. Linear Algebra Appl., 222:15-22, 1995. (p. 9).

[14] Stefan Neuwirth. Metric unconditionality and Fourier analysis. Studia Math., 131:19-62, 1998. (pp. 13, 18, 20, and 22).
[15] Stefan Neuwirth. *Multiplicateurs et analyse fonctionnelle*. PhD thesis, Université Paris 6, 1999. http://tel.archives-ouvertes.fr/tel-00010399. (p. 18).

[16] Stefan Neuwirth. The size of bipartite graphs with girth eight. http://arxiv.org/math/0102210, 2001. (p. 24).

[17] Vern I. Paulsen, Stephen C. Power, and Roger R. Smith. *Schur products and matrix completions*. *J. Funct. Anal.*, 85(1):151–178, 1989. (pp. 5 and 11).

[18] A. Pełczyński and P. Wojtaszczyk. *Banach spaces with finite dimensional expansions of identity and universal bases of finite dimensional subspaces*. *Studia Math.*, 40:91–108, 1971. (p. 19).

[19] Vladimir V. Peller. Hankel operators of class $\mathcal{S}_p$ and their applications (rational approximation, Gaussian processes, the problem of majorizing operators). *Math. USSR-Sb.*, 41(4):443–479, 1982. (p. 23).

[20] Vladimir V. Peller. *Hankel operators and their applications*. Springer Monographs in Mathematics. Springer-Verlag, New York, 2003. (p. 7).

[21] Gilles Pisier. *Non-commutative vector valued $L_p$-spaces and completely $p$-summing maps*. Number 247 in Astérisque. Société mathématique de France, 1998. (pp. 4, 5, 7, 10, and 23).

[22] Gilles Pisier. *Similarity problems and completely bounded maps*, volume 1618 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, expanded edition, 2001. Includes the solution to "The Halmos problem". (pp. 5, 7, and 11).

[23] Gilles Pisier and Quanhua Xu. Non-commutative $L^p$-spaces. In W. B. Johnson and J. Lindenstrauss, editors, *Handbook of the geometry of Banach spaces*, Vol. 2, pages 1459–1517. North-Holland, 2003. (pp. 9 and 19).

[24] Vojtěch Rödl. On a packing and covering problem. *European J. Combin.*, 6(1):69–78, 1985. (p. 18).

[25] Walter Rudin. *Fourier analysis on groups*. Interscience, 1962. (p. 9).

[26] Raymond A. Ryan. *Introduction to tensor products of Banach spaces*. Springer-Verlag, 2002. (p. 5).

[27] Carsten Schütt. Unconditionality in tensor products. *Israel J. Math.*, 31(3-4):209–216, 1978. (pp. 6 and 9).

[28] Josef A. Seigner. *Rademacher variables in connection with complex scalars*. *Acta Math. Univ. Comenian.* (N.S.), 66:329–336, 1997. (p. 9).

[29] H. van Maldeghem. *Generalized polygons*. Birkhäuser Verlag, 1998. (pp. 24 and 25).

[30] Nicholas Varopoulos. *Some combinatorial problems in harmonic analysis, lecture notes by D. Salinger from a course given by N. Th. Varopoulos at the Summer School in Harmonic Analysis, University of Warwick, 1st July-13th July, 1968*. Mathematics Institute, University of Warwick. (pp. 11 and 22).

[31] Nicholas Varopoulos. *Tensor algebras and harmonic analysis*. *Acta Math.*, 119:51–112, 1967. (p. 11).

[32] Nicholas Varopoulos. *Tensor algebras over discrete spaces*. *J. Functional Analysis*, 3:321–335, 1969. (pp. 9, 10, and 11).

*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.

2000 Mathematics Subject Classification. 47B10, 46B15, 46B04, 43A46, 05C38, 46B28.