Cycles and 1-unconditional matrices

Stefan Neuwirth

Abstract

We characterise the 1-unconditional subsets (e_{rc} )_{(r,c)∈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 ⩽ 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)∈I} in S^{p}.

Résumé en français

Je caractérise les sous-suites 1-inconditionnelles (e_{rc} )_{(r,c)∈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 ∈ {4, 6, …, 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)∈I} dans S^{p}.

1 Introduction

The starting point for this investigation has been the following isometric question on the Schattenvon-Neumann class S^{p}.

Question 1.1. Which matrix coefficients of an operator x ∈ 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 ⊆ R×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)∈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 ∈ S^{p} must vanish so that for all matrices φ of complex, or real, numbers

∥φ ∗ x∥ ⩽ sup|φ_{rc}| ∥x∥,

where φ ∗ x is the Schur (or Hadamard or entrywise) product defined by

(φ ∗ x)_{rc} = φ_{rc}x_{rc}?

In the case p = ∞, Grothendieck’s inequality yields an estimation for the norm of Schur mul-

tiplication by φ in terms of the projective tensor product ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} : this norm is equivalent to the

supremum of the norm of those elements of ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} whose coefficient matrices are finite submatrices of φ. 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 ∈ ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} 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∥ = (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 puples (v_{1}, v_{2}, …, v_{p}) satisfying (v_{2i−1} , v_{2i}), (v_{2i+1} , v_{2i}) ∈ 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 ∈ (0, ∞] \ {2, 4, 6, …}. If the sequence of elementary matrices (e_{rc})_{(r,c)∈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 φ ∈ ℓ_{I}^{∞} may be interpolated

by a tensor u ∈ ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} such that ∥u∥ = ∥φ∥.

Theorem 1.5. Let p ∈ {2, 4, 6, …}. The sequence (e_{rc})_{(r,c)∈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)∈I} is a 1-unconditional basic sequence in S^{p}, then we obtain the following hierarchy:

U_{p} for a p ∈ (0, ∞] \ {2, 4, 6, …} ⇒ ⋯⇒ U_{2n+2} ⇒ U_{2n} ⇒ ⋯⇒ 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 ∈ 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 ⩾ 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) ∈ ℤ/7ℤ × ℤ/7ℤ : r + c ∈ {0, 1, 3}}.  (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 1: The Heawood graph

Thus the p-trace norm of every matrix with pattern

( \begin{matrix}& 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ 0 & ∗ & ∗ & 0 & ∗ & 0 & 0 & 0 \\ 1 & ∗ & 0 & ∗ & 0 & 0 & 0 & ∗ \\ 2 & 0 & ∗ & 0 & 0 & 0 & ∗ & ∗ \\ 3 & ∗ & 0 & 0 & 0 & ∗ & ∗ & 0 \\ 4 & 0 & 0 & 0 & ∗ & ∗ & 0 & ∗ \\ 5 & 0 & 0 & ∗ & ∗ & 0 & ∗ & 0 \\ 6 & 0 & ∗ & ∗ & 0 & ∗ & 0 & 0 \end{matrix} )

does not depend on the sign of its coefficients if and only if p ∈ {2, 4}.

These results give a complete description of the situation in which (e_{rc})_{(r,c)∈I} is a 1-uncondi-tional 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 ∈ [1, ∞] \ {2, 4, 6, …}. 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 ∈ {2, 4, 6, …}. The space S_{I}^{p} has complex, or real, (muap) if and only if any two vertices at distance 2j + 1 ⩽ 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 ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} 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 p-trace norm and closed walks of length p 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 n-independent subsets of a discrete abelian group, Hankel sets, Steiner systems and Tits’ 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 ⨿ R: if there is a risk of confusion, an element n ∈ 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} ⊆ 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 ∈ C and r ∈ R and may therefore be given alternatively by the set of couples I = {(r, c) ∈ R × C : {r, c} ∈ 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 × 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) ∈ I ⇒ r = r′, and a row section if (r, c), (r, c′) ∈ I ⇒ c = c′.

A walk of length s ⩾ 0 in a graph is a sequence (v_{0}, …, v_{s}) of s + 1 vertices such that {v_{0}, v_{1}}, …, {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 ⩾ 0 in a graph is a sequence (v_{1}, …, v_{p}) of p vertices such that {v_{1}, v_{2}}, …, {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 ⨿R is nonempty, then its first vertex is a column vertex: v_{1} ∈ 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 ∅. If v is a vertex and (r, c, …, v) is the unique path from r to v, let the word c^{⌢} ⋯^{⌢}v index v. Let W be the set of all words thus formed. Then

• ∅ ∈ W and every beginning of a word in W is also in W: if w ∈ W \ {∅}, then w is the concatenation w′^{⌢}v of a word w′ ∈ 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^{⌢}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 𝕋 = {z ∈ ℂ : |z| = 1}.

The unit ball of a Banach space X is denoted by B_{X}.

Given an index set I and q ∈ I, e_{q} is the sequence defined on I as the indicator function χ_{{q}} of the singleton {q}.

Let I = R × C and q = (r, c). Then e_{q} = e_{rc} is the elementary matrix identified with the operator from ℓ_{C}^{2} to ℓ_{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 ℓ_{C}^{2} to ℓ_{R}^{2} is x_{q} = tr e_{q}^{∗}x and its matrix representation is (x_{q})_{q∈R×C} = ∑ _{q∈R×C} x_{q}e_{q}. The support of x is {q ∈ R × C : x_{q} ≠ 0}.

The Schatten-von-Neumann class S^{p}, 0 < p < ∞, is the space of those compact operators x from ℓ_{C}^{2} to ℓ_{R}^{2} such that ∥x∥_{p}^{p} = tr|x|^{p} = tr(x^{∗}x)^{p/2} < ∞. S^{∞} is the space of compact operators with the operator norm. S^{p} is a quasi-normed space, and a Banach space if p ⩾ 1. Let (R_{n} × C_{n})_{n⩾0} be a sequence of finite sets that tends to R × C. Then the sequence of operators P_{n} : x ↦ ∑ _{q∈R_{n} ×C_{n}} x_{q}e_{q} tends pointwise to the identity on S^{p} if p ⩾ 1.

For I ⊆ R × C, the entry space S_{I}^{p} is the subspace of those x ∈ 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∈I} .

The S^{p}-valued Schatten-von-Neumann class S^{p}(S^{p}) is the space of those compact operators x from ℓ_{C}^{2} to ℓ_{R}^{2}(S^{p}) such that ∥x∥_{p}^{p} = tr(tr|x|^{p}) < ∞, 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 ∈ S^{p}(S^{p}) can also be considered as a compact operator from ℓ_{C}^{2}(ℓ_{2}) = ℓ_{2} ⊗_{2} ℓ_{C}^{2} to ℓ_{R}^{2}(ℓ_{2}) = ℓ_{2} ⊗_{2} ℓ_{R}^{2} such that ∥x∥_{p}^{p} = tr⊗ tr|x|^{p} < ∞; the matrix coefficient of x at q is then x_{q} = (Id_{S^{p}} ⊗ tr)((Id_{ℓ_{2}} ⊗ e_{q}^{∗})x) and its matrix representation is ∑ _{q∈R×C} x_{q} ⊗ 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 φ = (φ_{q})_{q∈I} ∈ ℂ^{I} such that the associated Schur multiplication operator M_{φ} defined by e_{q} ↦ φ_{q}e_{q} for q ∈ I is bounded on S_{I}^{p}. The Schur multiplier φ is furthermore completely bounded (c.b. for short) on S_{I}^{p} if Id_{S^{p}} ⊗ M_{φ}, the operator defined by x_{q}e_{q} ↦ φ_{q}x_{q}e_{q} for x_{q} ∈ S^{p} and q ∈ I, is bounded on S_{I}^{p}(S^{p}) (see [21, Lemma 1.7].) The norm of φ is the norm of M_{φ} and its complete norm is the norm of Id_{S^{p}} ⊗ M_{φ}. This norm is the supremum of the norm of its restrictions to finite rectangle sets R′× C′. Note that φ is a Schur multiplier on S^{∞} if and only if, for every bounded operator x: ℓ_{C}^{2} → ℓ_{R}^{2}, (φ_{q}x_{q}) is the matrix representation of a bounded operator; also φ is automatically c.b. on S^{∞} [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 Γ = Ĝ be the dual group of characters on G. The spectrum of an integrable function f on G is {γ ∈ Γ : f̂(γ) ≠ 0}. Let Λ ⊆ Γ. If X is a space of integrable functions on G, then X_{Λ} is the translation invariant subspace of those f ∈ X whose spectrum is a subset of Λ.

Let X be the space of continuous functions C(G) or the Lebesgue space L^{p}(G) with 0 < p < ∞. Then X_{Λ} is also the closed subspace of X spanned by Λ. A relative Fourier multiplier on X_{Λ} is a sequence μ = (μ_{γ})_{γ∈Λ} ∈ ℂ^{Λ} such that the associated convolution operator M_{μ} defined by γ ↦ μ_{γ}γ for γ ∈ Λ is bounded on X_{Λ} . The Fourier multiplier μ is furthermore c.b. if Id_{S^{p}} ⊗ M_{μ}, the operator defined by a_{γ}γ ↦ μ_{γ}a_{γ}γ for a_{γ} ∈ S^{p} and γ ∈ Λ, is bounded on the S^{p}-valued space X_{Λ} (S^{p}) (where p = ∞ if X = C(G).) The norm of μ is the norm of M_{μ} and its complete norm is the norm of Id_{S^{p}} ⊗ M_{μ}. Note that μ is a Fourier multiplier on C_{Λ} (G) if and only if, for every f ∈ L_{Λ}^{∞} (G), ∑ μ_{γ} f̂(γ)γ is the Fourier series of an element of L_{Λ}^{∞} (G): μ is a relative Fourier multiplier on L^{∞} (G); also μ is automatically c.b. on C_{Λ} (G) [22, Cor. 3.18].

Let X, Y be Banach spaces and u ∈ X ⊗ Y. Its projective tensor norm is

∥u∥_{X ⊗^{∧} Y} = inf{ ∑_{j=1}^{n} ∥x_{j} ∥ ∥y_{j} ∥ : u = ∑_{j=1}^{n} x_{j} ⊗ y_{j} }

and X ⊗^{∧} Y is the completion of X ⊗ Y with respect to this norm. Note that ℓ_{∞}^{n} ⊗^{∧} ℓ_{∞}^{m} ⊂ c_{0} ⊗^{∧} c_{0}

because ℓ_{∞}^{n} and ℓ_{∞}^{m} are 1-complemented in c_{0}, and that c_{0} ⊗^{∧} c_{0} ⊂ ℓ_{∞} ⊗^{∧} ℓ_{∞} because ℓ_{∞} is the bidual of c_{0}.

Let ∑ x_{j} ⊗ y_{j} be any representation of the tensor u. If ξ ⊗ η ∈ X^{∗} ⊗ Y^{∗}, we define ⟨ξ ⊗ η, u⟩ = ∑ ⟨ξ, x_{j} ⟩⟨η, y_{j} ⟩. The injective tensor norm of u is

∥u∥_{X ⊗^{∨} Y} = sup_{(ξ,η)∈B_{X^{∗}} ×B_{Y ^{∗}}} |⟨ξ ⊗ η, u⟩|

and X ⊗^{∨} Y is the completion of X ⊗ Y with respect to this norm.

If X and Y are both finite dimensional, then

(X ⊗^{∨} Y )^{∗} = X^{∗} ⊗^{∧} Y^{∗} and (X ⊗^{∧} Y )^{∗} = X^{∗} ⊗^{∨} Y^{∗}.

Further (c_{0} ⊗^{∧} c_{0})^{∗} = ℓ_{1} ⊗^{∨} ℓ_{1}: in fact, (c_{0} ⊗^{∧} c_{0})^{∗} may be identified with the space of bounded operators

from c_{0} to ℓ_{1} and ℓ_{1} ⊗^{∨} ℓ_{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 ℓ_{1} is compact and ℓ_{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 ⟨e_{c} ⊗ e_{r}, u⟩. 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 ⊆ R × C and φ be a Schur multiplier on S_{I}^{∞} with norm D. Then φ is also a c.b. Schur multiplier on S_{I}^{p} for every p ∈ (0, ∞], with complete norm bounded by D.

Proof. We may assume that D = 1. Let R′× C′ be any finite subset of R × 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 φ_{rc} = ⟨w_{c}, v_{r}⟩ for every (r, c) ∈ I ∩ R′× C′. If we define W: ℓ_{C′}^{2} → ℓ_{C′}^{2}(H) and V : ℓ_{R′}^{2} → ℓ_{R′}^{2}(H) by W ζ = (ζ_{c}w_{c})_{c∈C′} and V η = (η_{r}v_{r})_{r∈R′} , then V and W have norm at most 1, and the proposition follows from the factorisation

M_{φ} x = V^{∗}(x ⊗ Id_{H})W

for every x with support in I ∩ R′× C′.

Remark 2.2. Éric Ricard showed us an elementary proof that a Schur multiplier on S_{I}^{∞} is automatically c.b., included here by his kind permission. A Schur multiplier φ is bounded on S_{I}^{∞} by a constant D if and only if

∀ ξ ∈ B_{S_{I}^{∞}} ∀ η ∈ B_{ℓ_{R}^{2}} ∀ ζ ∈ B_{ℓ_{C}^{2}}  | ∑_{(r,c)∈I} η_{r}φ_{rc}ξ_{rc}ζ_{c} | ⩽ D.  (2)

It is furthermore completely bounded on S_{I}^{∞} by D if

∀ x ∈ B_{S_{I}^{∞} (S^{∞} )} ∀ y ∈ B_{ℓ_{R}^{2} (ℓ_{2} )} ∀ z ∈ B_{ℓ_{C}^{2} (ℓ_{2} )} | ∑_{(r,c)∈I} φ_{rc}⟨y_{r}, x_{rc}z_{c}⟩| ⩽ D.  (3)

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

ξ_{rc} = ⟨y_{r}/∥y_{r}∥, x_{rc}z_{c}/∥z_{c}∥⟩, η_{r} = ∥y_{r}∥_{ℓ_{2}} and ζ_{c} = ∥z_{c}∥_{ℓ_{2}}.

Then ∥η∥_{ℓ_{R}^{2}} , ∥ζ∥_{ℓ_{C}^{2}} ⩽ 1 and

∥ξ∥ = sup{ | ∑_{(r,c)∈I} ⟨α_{r}y_{r}/∥y_{r}∥_{ℓ_{2}}, x_{rc}β_{c}z_{c}/∥z_{c}∥_{ℓ_{2}}⟩| : α ∈ B_{ℓ_{R}^{2}} , β ∈ B_{ℓ_{C}^{2}} }

⩽ ∥x∥ sup_{α∈B_{ℓ{2 \atop R}}} ∥(α_{r}y_{r}/∥y_{r}∥_{ℓ_{2}})∥_{ℓ_{R}^{2} (ℓ_{2} )} sup_{β∈B_{ℓ{2 \atop C}}} ∥(β_{c}z_{c}/∥z_{c}∥_{ℓ_{2}})∥_{ℓ_{C}^{2} (ℓ_{2} )} ⩽ 1,

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

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

∀ ζ ∈ 𝕋^{C} ∀ η ∈ 𝕋^{R} ∥ ∑_{(r,c)∈R×C} ζ_{c}η_{r}a_{rc} e_{rc} ∥_{p} = ∥ ∑_{(r,c)∈R×C} a_{rc} e_{rc} ∥_{p}  (4)

for every finitely supported sequence of complex or S^{p}-valued coefficients a_{rc}. Let ζ ⊗ η denote the elementary Schur multiplier (ζ_{c}η_{r})_{(r,c)∈R×C} . Equation (4) shows that if ζ ∈ 𝕋^{C} and η ∈ 𝕋^{R}, then M_{ζ⊗η} is an isometry on every S^{p}. This yields that if ζ ∈ ℓ_{C}^{∞} , η ∈ ℓ_{R}^{∞} , then the complete norm of M_{ζ⊗η} is ∥ζ∥_{ℓ_{C}^{∞}} ∥η∥_{ℓ_{R}^{∞}} 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} → S_{I}^{p} be an operator. Let R′× C′ be a finite subset of R × C and let P_{R′×C′} be the contractive projection onto S_{R′×C′}^{p} defined by the Schur multiplier χ_{C′} ⊗ χ_{R′}. Then the average of T with respect to R′× C′ is given by

[T]_{R′×C′} (x) = ∫_{𝕋^{R}} dη ∫_{𝕋^{C}} dζ M_{ζ^{∗} ⊗η^{∗}} P_{R′×C′} T (M_{ζ⊗η} x),  (5)

where ζ^{∗} = (\overline{ζ_{c}})_{c∈C} and η^{∗} = (\overline{η_{r}})_{r∈R} .

Proposition 2.4. Let T: S_{J}^{p} → S_{I}^{p} be an operator and R′ × C′ a finite subset of R × C. Then [T]_{R′×C′} is a Schur multiplication operator from S_{J}^{p} to S_{I∩R′×C′}^{p} such that ∥[T]_{R′×C′} ∥ ⩽ ∥T ∥. In fact, [T]_{R′×C′} = M_{φ^{R′×C′}} with

φ_{rc}^{R′×C′} = \{ \begin{matrix}tr e_{rc}^{∗}T (e_{rc}) & if (r, c) ∈ J ∩ R′× C′ \\ 0 & if (r, c) ∈ J \ R′× C′. \end{matrix}

If T is a projection onto S_{I}^{p}, then φ^{R′×C′} = χ_{I∩R′×C′} , so that [T]_{R′×C′} is a projection onto S_{I∩R′×C′}^{p} . Let φ = (tr e_{q}^{∗}T (e_{q}))_{q∈J} . Then ∥M_{φ} ∥ ⩽ ∥T ∥ and we define the average of T by [T] = M_{φ}.

Proof. Formula (5) shows that ∥[T]_{R′×C′} (x)∥ ⩽ ∥T ∥ ∥x∥. We have

[T]_{R′×C′} (e_{rc})= ∫_{𝕋^{R}} dη ∫_{𝕋^{C}} dζ M_{ζ^{∗} ⊗η^{∗}} P_{R′×C′} T (ζ_{c}η_{r} e_{rc})

= ∫_{𝕋^{R}} dη ∫_{𝕋^{C}} dζ ζ_{c}η_{r}M_{ζ^{∗} ⊗η^{∗}} ∑_{(r′,c′)∈R′×C′} tr(e_{r′c′}^{∗} T (e_{rc}))e_{r′c′}

= ∑_{(r′,c′)∈R′×C′} ∫_{𝕋^{R}} dη ∫_{𝕋^{C}} dζ ζ_{c}η_{r} tr(e_{r′c′}^{∗} T (e_{rc}))ζ_{c′}^{−1} η_{r′}^{−1} e_{r′c′} = φ_{rc}^{R′×C′} e_{rc}.

As the norm of a Schur multiplier is the supremum of the norm of its restrictions to finite rectangle sets, this shows that φ is a Schur multiplier on S_{J}^{p} and ∥M_{φ} ∥ ⩽ ∥T ∥. If T is a projection onto S_{I}^{p}, note that tr e_{rc}^{∗}T (e_{rc}) = χ_{I} (r, c).

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 Γ be a countable discrete abelian group and Λ ⊆ Γ. Let R and C be two copies of Γ and consider I = {(r, c) ∈ R × C : r − c ∈ Λ}. Let φ ∈ ℂ^{I} such that there is μ ∈ ℂ^{Λ} with φ(r, c) = μ(r − c) for all (r, c) ∈ I. Let G = Γ̂, so that Γ is the group of characters on the compact group G. Let p ∈ (0, ∞].

(a) The complete norm of the relative Schur multiplier φ on S_{I}^{p} is bounded by the complete norm of the relative Fourier multiplier μ on L_{Λ}^{p} (G).

(b) Suppose that Γ is finite. The norm of the relative Fourier multiplier μ on L_{Λ}^{p} (G) is bounded by the norm of the relative Schur multiplier φ 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 Γ = ℤ/nℤ is treated.

Proof. (a) is [21, Lemma 8.1.4]: for all a_{q} ∈ S^{p}, of which only a finite number are nonzero, and all g ∈ G, we have by matrix unconditionality (Eq. (4))

∥∑_{q∈I} a_{q}e_{q}∥_{S_{I}^{p} (S^{p})} = ∥ ∑_{(r,c)∈I} r(g)c(g)^{−1} a_{rc} e_{rc} ∥_{S_{I}^{p} (S^{p})}

= ∥∑_{γ∈Λ} ( ∑_{\begin{matrix}(r,c)∈I \\ r−c=γ \end{matrix}} a_{rc} e_{rc}) γ(g)∥_{S_{I}^{p} (S^{p})} = ∥∑_{γ∈Λ} ( ∑_{\begin{matrix}(r,c)∈I \\ r−c=γ \end{matrix}} a_{rc} e_{rc}) γ∥_{L_{Λ}^{p} (G,S^{p} (S^{p}))} . (6)

This yields an isometric embedding of S_{I}^{p}(S^{p}) in L_{Λ}^{p} (G, S_{I}^{p}(S^{p})). As S^{p}(S^{p}) may be identified with S^{p}(ℓ_{Γ}^{2}(ℓ^{2})),

∥∑_{q∈I} φ_{q}a_{q}e_{q}∥_{S_{I}^{p} (S^{p})} = ∥∑_{γ∈Λ} μ_{γ}( ∑_{\begin{matrix}(r,c)∈I \\ r−c=γ \end{matrix}} a_{rc} e_{rc}) γ∥_{L_{Λ}^{p} (G,S^{p} (S^{p}))} ⩽ ∥Id ⊗ M_{μ}∥∥∑_{q∈I} a_{q}e_{q}∥_{S_{I}^{p} (S^{p})} .

(b). Let us embed L_{Λ}^{p} (G) into S_{I}^{p} by f ↦ mf̂, where mf̂: ℓ_{C}^{2} → ℓ_{R}^{2} is the convolution operator defined by

mf̂e_{c} = f̂∗ e_{c} = ∑_{γ∈Λ} f̂(γ)e_{γ} ∗ e_{c} = ∑_{r−c∈Λ} f̂(r − c)e_{r} :

mf̂ has the matrix representation ∑ _{(r,c)∈I} f̂(r − c)e_{rc}. The characters g ∈ G form an orthonormal basis for ℓ_{C}^{2} such that mf̂g = f(g)g: therefore

∥mf̂∥_{p} =( ∑_{g∈G} |f(g)|^{p})^{1/p} = (# G)^{1/p}∥f∥_{L^{p} (G)} .

As M_{φ} mf̂ = m{ \atop M_{μ} f}, this shows that the norm of μ on L_{Λ}^{p} (G) is the norm of φ on the subspace of circulant matrices in S_{I}^{p}. The same holds for complete norms.

3 Idempotent Schur multipliers of norm 1

A Schur multiplier is idempotent if it is the indicator function χ_{I} of some set I ⊆ R × C; if χ_{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 ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} with 0, 1 coefficients of norm 1 may be characterised by the combinatorics of I.

Proposition 3.1. Let I ⊆ R × C be nonempty and 0 < p ≠ 2 < ∞. The following are equivalent.

(a) For every finite rectangle set R′× C′ intersecting I

∥ ∑_{(r,c)∈I∩R′×C′} e_{c} ⊗ e_{r} ∥_{ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞}} = 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} ⊆ R and pairwise disjoint sets C_{j} ⊆ C such that I = ⋃ R_{j} × C_{j}.

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

( \begin{matrix}& C_{1} & C_{2} & C_{3} & ··· \\ R_{1} & ∗ & 0 & 0 & ⋯ \\ R_{2} & 0 & ∗ & 0 & ... \\ R_{3} & 0 & 0 & ∗ & ... \\ ⋮ & ⋮ & ... & ... & ... \end{matrix}) .

Proof. (b) ⇒ (c) is trivial.

(a) ⇒ (b). The complete norm of a Schur multiplier φ on S^{p} is the supremum of the complete norm of its restrictions φ′ = (φ_{q})_{q∈R′×C′} to finite rectangle sets R′× C′. Furthermore, the complete norm of an elementary Schur multiplier (η_{c}ζ_{r})_{(r,c)∈R×C} = η ⊗ ζ on S^{p} equals ∥η∥_{ℓ_{C}^{∞}} ∥ζ∥_{ℓ_{R}^{∞}} .

(c) ⇒ (d). If I is not a union of pairwise disjoint complete bipartite graphs, then there are r_{0}, r_{1} ∈ R and c_{0}, c_{1} ∈ C such that

I′ = I ∩ {r_{0}, r_{1}} × {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}} × {c_{0}, c_{1}} would be the contractive projection associated to the Schur multiplier χ_{I′}. Let x(t), t ∈ ℝ, be the operator from ℓ_{C}^{2} to ℓ_{R}^{2} whose matrix coefficients vanish except for its {r_{0}, r_{1}}×{c_{0}, c_{1}} submatrix,

which is ( \begin{matrix}1 & \sqrt{2} \\ \sqrt{2} & t \end{matrix}) . Its eigenvalues are

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

so that

\{ \begin{matrix}∥x(t)∥_{∞} = 2 + t/3 + o(t) \\ ∥x(t)∥_{p}^{p} = 2^{p} + 1 + p(2^{p} − 4)t/6 + o(t) for 0 < p < ∞ \end{matrix}

and therefore ∥χ_{I′} ∗ x(t)∥_{p} = ∥x(0)∥_{p} > ∥x(t)∥_{p} for some t ≠ 0 if p ≠ 2.

(d) ⇒ (a). Suppose (d) and let R′× C′ intersect I. Then there are pairwise disjoint sets R′_{j} and pairwise disjoint sets C′_{j} such that I ∩ R′× C′ = R′_{1} × C′_{1} ∪ ⋯∪ R′_{n} × C′_{n} and

∑_{(r,c)∈I∩R′×C′} e_{c} ⊗ e_{r} = ∑_{j=1}^{n} χ_{C′_{j}} ⊗ χ_{R′_{j}} = average_{ϵ_{j} =±1}( ∑_{j=1}^{n} ϵ_{j} χ_{C′_{j}}) ⊗ ( ∑_{j=1}^{n} ϵ_{j} χ_{R′_{j}} )

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_{χ_{I}} : S^{∞} → S_{I}^{∞} is either 1 or at least 2/\sqrt{3}, as

∥( \begin{matrix}1 & \sqrt{2} \\ \sqrt{2} & −1 \end{matrix}) ∥_{∞} = \sqrt{3},  ∥( \begin{matrix}1 & \sqrt{2} \\ \sqrt{2} & 0 \end{matrix}) ∥_{∞} = 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_{χ_{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} ⊗ e_{0} + e_{0} ⊗ e_{1} + e_{1} ⊗ e_{0} =

((e^{−iπ/12} , e^{iπ/4} ) ⊗ (e^{−iπ/12} , e^{iπ/4} ) + (e^{iπ/12} , e^{−iπ/4} ) ⊗ (e^{iπ/12} , e^{−iπ/4} ))/\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 ⩽ ∞ and I ⊆ R × C. Let 𝕊 = 𝕋 (vs. 𝕊 = {−1, 1}.)

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

∥∑_{q∈I} ϵ_{q}a_{q}e_{q}∥_{p} ⩽ D∥∑_{q∈I} a_{q}e_{q}∥_{p}  (7)

for every choice of signs ϵ_{q} ∈ 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 ϵ_{q} ∈ 𝕊 and every finitely supported sequence of operator coefficients a_{q} ∈ 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

∥∑_{q∈I} ϵ_{q}a_{q}e_{q}∥_{p} = ∥∑_{q∈I} a_{q}e_{q}∥_{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 Dπ/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 σ(p) sets and complete σ(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 × {c}, a single row {r} × C, the diagonal set {(rown, coln)}_{n∈ℕ} if R and C are copies of 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 ⊆ R × C and 0 < p ⩽ ∞. If I is a real (vs. complex) 1-unconditional basic sequence in S^{∞} , then I is also a real (vs. complex) completely 1-unconditional basic sequence in S^{p}.

Example 4.4. If R = C = {0, …, n − 1}, 1 ⩽ p ⩽ ∞ and I = R × 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 kth tensor power ( \begin{matrix}1 & 1 \\ 1 & −1 \end{matrix} ) ^{⊗k} (the kth 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^{∞} is 5/3 and differs from its complex unconditional constant, \sqrt{3}. In fact, because the canonical bases of ℓ_{C}^{2} and ℓ_{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{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}) , ( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & −1 \end{matrix} ) or ( \begin{matrix}−1 & 1 & 1 \\ 1 & −1 & 1 \\ 1 & 1 & −1 \end{matrix} ) .

The first one has norm 1: it defines the identity. The second one has the same norm as the Schur

multiplier ( \begin{matrix}1 & 1 \\ 1 & −1 \end{matrix} ) , which is \sqrt{2}, because the norm of that multiplier equals the norm of its tensor

product by Id_{ℓ_{2}^{2}}, which is ( \begin{matrix}1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & −1 & −1 \\ 1 & 1 & −1 & −1 \end{matrix} ) . By Prop. 2.5 for Γ = ℤ/3ℤ, the third one has the

same norm as the Fourier multiplier φ = (−1, 1, 1) on L^{∞} (G), where G = {z ∈ 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

(| − 1 + 1 + 1| + | − 1 + e^{2iπ/3} + e^{4iπ/3} | + | − 1 + e^{4iπ/3} + e^{2iπ/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 ≠ 2.

5 Varopoulos’ characterisation of unconditional matrices in S^{∞}

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^{∞} in his study of the projective tensor product c_{0} ⊗^{∧} 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 ⊆ R × C. The following are equivalent.

(a) I is an unconditional basic sequence in S^{∞} .

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

(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) ⊗^{∧} c_{0}(R).

(d) The coefficients of every tensor in ℓ_{C}^{1} ⊗^{∨} ℓ_{R}^{1} with support in I form an absolutely convergent series.

(e) (z_{c}z′_{r})_{(r,c)∈I} is a Sidon set in the dual of 𝕋^{C} × 𝕋^{R}, that is, an unconditional basic sequence in C(𝕋^{C} × 𝕋^{R}).

(f) There is a constant λ such that for all R′ ⊆ R and C′ ⊆ C with n elements # [I ∩R′×C′] ⩽ λn.

(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^{∞} .

Sketch of proof. (a) ⇒ (b). If (a) holds, every sequence of signs ϵ ∈ {−1, 1}^{I} is a Schur multiplier on S_{I}^{∞} . By a convexity argument, this implies that every bounded sequence is a Schur multiplier on S_{I}^{∞} , which may be extended to a Schur multiplier on S^{∞} 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

∥ ∑_{(r,c)∈I} a_{rc} e_{c} ⊗ e_{r} ∥_{ℓ_{C}^{1} ⊗^{∨} ℓ_{R}^{1}} = sup_{|z_{c} |,|z′_{r}|=1} | ∑_{(r,c)∈I} a_{rc}z_{c}z′_{r}|.  (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^{⌢}r, w) with w a word and r 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^{∞} and are 1-complemented in S^{∞} 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

Φ_{I} (ϵ, a) = tr|∑_{q∈I} ϵ_{q}a_{q}e_{q}|^{p}  (9)

for I ⊆ R × C, a positive even integer p = 2k, signs ϵ_{q} ∈ 𝕋 and coefficients a_{q} ∈ ℂ, of which only a finite number are nonzero. In fact,

Φ_{I} (ϵ, a) = tr( ∑_{(r,c),(r′,c′)∈I} (ϵ_{rc}a_{rc} e_{rc})^{∗}(ϵ_{r′c′} a_{r′c′} e_{r′c′}))^{k}

= tr ∑_{\begin{matrix}(r_{1} ,c_{1} ),(r′_{1},c′_{1} ),..., \\ (r_{k} ,c_{k}),(r′_{k} ,c′_{k})∈I \end{matrix}} ∏_{i=1}^{k} (ϵ_{r_{i} c_{i}}^{−1} \overline{a_{r_{i} c_{i}}} e_{c_{i} r_{i}} )(ϵ_{r′_{i}c′_{i}} a_{r′_{i}c′_{i}} e_{r′_{i}c′_{i}} )

= ∑_{\begin{matrix}(r_{1} ,c_{1} ),(r_{1} ,c_{2} ),..., \\ (r_{k} ,c_{k}),(r_{k} ,c_{k+1} )∈I \end{matrix}}  ∏_{i=1}^{k} ϵ_{r_{i} c_{i}}^{−1} ϵ_{r_{i} c_{i+1}} \overline{a_{r_{i} c_{i}}} a_{r_{i} c_{i+1}} (where c_{k+1} = c_{1}.)

(10)

The latter sum runs over all closed walks (c_{1}, r_{1}, c_{2}, …, c_{k}, r_{k}) of length p in the graph I. With multinomial notation, its terms have the form ϵ^{β−α} \overline{a}^{α} a^{β} with |α| = |β| = k. The attempt to describe those couples (α, β) that effectively arise in this expansion yields the following definition.

Definition 6.1. Let p = 2k ⩾ 0 be an even integer and I ⊆ R × C.

(a) Let A_{k}^{I} = {α ∈ ℕ^{I} : ∑ _{q∈I} α_{q} = k} and set

B_{k}^{I} = {(α, β) ∈ A_{k}^{I} × A_{k}^{I} : ∀ r ∑ _{c} α_{rc} = ∑ _{c} β_{rc} and ∀ c ∑ _{r} α_{rc} = ∑ _{r} β_{rc}}.

(b) Two couples (α^{1}, β^{1}) ∈ B_{k_{1}}^{I}, (α^{2}, β^{2}) ∈ B_{k_{2}}^{I} are disjoint if k_{1}, k_{2} ⩾ 1 and

α_{rc}^{1} ⩾ 1 ⇒ ∀ (r′, c) ∈ I α_{r′c}^{2} = 0 and ∀ (r, c′) ∈ I α_{rc′}^{2} = 0.  (11)

(c) The set 𝒲_{k}^{I} of closed walk relations of length p in I is the subset of those (α, β) ∈ 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}, …, c_{k}, r_{k}) of length p we associate the couple (α, β) ∈ A_{k}^{I} × A_{k}^{I} defined by

α_{q} = # [i ∈ {1, …, k} : (r_{i}, c_{i}) = q]
β_{q} = # [i ∈ {1, …, k} : (r_{i}, c_{i+1} ) = q] (where c_{k+1} = c_{1}.)

We shall write P ∼ (α, β) and call n_{αβ} the number of elements of W_{k}^{I} mapped onto (α, β).

Note that the conditions in Eq. (11) is in fact symmetric and that it may be stated with β^{1} and β^{2} instead of α^{1} and α^{2}.

Example 6.2. Let R = C = {0, 1, 2, 3} and I = R × 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} \ 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 × C = {0, 1} × {0, 1}. Two closed walks are associated with the closed walk relation (e_{00} + e_{11}, e_{01} + e_{10}) ∈ 𝒲_{2}^{I}: the two cycles (col0, row0, col1, row1) and (col1, row1, col0, row0). 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 (col1, row0), (col1, row0), (col0, row0), (col0, row0).

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

Proposition 6.4. Let p = 2k ⩾ 0 be an even integer and I ⊆ R × C. The image of the mapping in Def. 6.1(d) is 𝒲_{k}^{I}:

(a) if P ∈ W_{k}^{I} and P ∼ (α, β), then (α, β) ∈ 𝒲_{k}^{I};

(b) if (α, β) ∈ 𝒲_{k}^{I}, then there is a P ∈ W_{k}^{I} such that P ∼ (α, β), so that n_{αβ} ⩾ 1.

Proof. (a). Let P = (c_{1}, r_{1}, c_{2}, r_{2}, …, c_{k}, r_{k}). In fact,

∑ _{c} α_{rc} = # [i ∈ {1, …, k} : r_{i} = r] = ∑ _{c} β_{rc}
∑ _{r} α_{rc} = # [i ∈ {1, …, k} : c_{i} = c] = ∑ _{r} β_{rc}

and (α, β) ∈ B_{k}^{I}. If (α, β) = (α^{1}, β^{1}) + (α^{2}, β^{2}) with (α^{i}, β^{i}) ∈ B_{k_{i}}^{I} and k_{i} ⩾ 1, there is an i such that α_{r_{i} c_{i}}^{1} ⩾ 1 and α_{r_{i+1} c_{i+1}}^{2} ⩾ 1 (where (r_{i+1} , c_{i+1} ) = (r_{1}, c_{1}) if i = k.) If β_{r_{i} c_{i+1}}^{1} ⩾ 1, then ∑ _{r} α_{rc_{i+1}}^{1} ⩾ 1, so that there is an r such that α_{rc_{i+1}}^{1} ⩾ 1. Otherwise β_{r_{i} c_{i+1}}^{2} ⩾ 1, so that ∑ _{c} α_{r_{i} c}^{2} ⩾ 1 and there is a c such that α_{r_{i} c}^{2} ⩾ 1. Therefore (α^{1}, β^{1}) and (α^{2}, β^{2}) are not disjoint and (α, β) ∈ 𝒲_{k}^{I}.

(b). We have to find a closed walk of length p that is mapped onto (α, β). If k = 0, the empty closed walk suits. Suppose that k ⩾ 1; Consider a walk (c_{1}, r_{1}, c_{2}, r_{2}, …, c_{j} , r_{j} , c_{j+1} ) in I such that α_{q}^{1} = # [i : (r_{i}, c_{i}) = q] ⩽ α_{q} and β_{q}^{1} = # [i : (r_{i}, c_{i+1} ) = q] ⩽ β_{q} for every q ∈ R × C, and furthermore j is maximal. We claim (A) that c_{j+1} = c_{1} and (B) that j = k. Let (α^{2}, β^{2}) = (α, β) − (α^{1}, β^{1}).

(A). If c_{j+1} ≠ c_{1}, then

∑ _{r} α_{rc_{j+1}}^{1} = # [i ∈ {1, …, j} : c_{i} = c_{j+1} ]
∑ _{r} β_{rc_{j+1}}^{1} = # [i ∈ {1, …, j + 1} : c_{i} = c_{j+1} ] = 1 + ∑ _{r} α_{rc_{j+1}}^{1} ,

so that there must be r_{j+1} with α_{r_{j+1} c_{j+1}}^{2} ⩾ 1. But then

∑ _{c} β_{r_{j+1} c}^{2} = ∑ _{c} α_{r_{j+1} c}^{2} ⩾ 1

and there must be c_{j+2} such that β_{r_{j+1} c_{j+2}}^{2} ⩾ 1: j is not maximal.

(B). Suppose that j < k. Then (α^{1}, β^{1}) ∈ B_{j}^{I} and (α^{2}, β^{2}) ∈ B_{k−j}^{I} . By hypothesis, they are not disjoint: there are r, c, c′ such that α_{rc}^{1}α_{rc′}^{2} ⩾ 1 or r, r′, c such that α_{rc}^{1}α_{r′c}^{2} ⩾ 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 α_{r′_{1}c′_{1}}^{2} ⩾ 1. Then there is c′_{2} such that β_{r′_{1}c′_{2}}^{2} ⩾ 1. By the argument used in Claim (A), there is a closed walk (c′_{1}, r′_{1}, c′_{2}, …, c′_{j′}, r′_{j′}) such that # [i : (r′_{i}, c′_{i}) = q] ⩽ α_{q}^{2} and # [i : (r′_{i}, c′_{i+1} ) = q] ⩽ β_{q}^{2} (where c′_{j′+1} = c′_{1}.) Then the closed walk

(c_{1}, r_{1}, c_{2}, r_{2}, …, c_{j} , r_{j} , c′_{2}, r′_{2}, …, c′_{j′}, r′_{j′}, c′_{1}, r′_{1})

shows that j is not maximal.

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 ⊆ R × C.

(a) The function Φ_{I} in Eq. (9) has the expansion

Φ_{I} (ϵ, a) = ∑_{(α,β)∈𝒲_{k}^{I}} n_{αβ} ϵ^{β−α} \overline{a}^{α} a^{β},  (12)

where n_{αβ} ⩾ 1 for every (α, β) ∈ 𝒲_{k}^{I}.

(b) If ϵ ∈ 𝕋^{I} and a ∈ (S^{p})^{I} is finitely supported, then the function

Ψ_{I} (ϵ, a) = tr|∑_{q∈I} ϵ_{q}a_{q}e_{q}|^{p}  (13)

has the expansion

∑_{(α,β)∈𝒲_{k}^{I}} ϵ^{β−α} ∑_{(c_{1} ,r_{1} ,...,c_{k} ,r_{k})∼(α,β)} ∏_{i=1}^{k} a_{r_{i} c_{i}}^{∗} a_{r_{i} c_{i+1}} (with c_{k+1} = c_{1}.)  (14)

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

Note that the edges of a closed walk P ∼ (α, β) are precisely those {r, c} such that α_{rc} + β_{rc} ⩾ 1. P is a cycle if and only if P does not have length 0 or 2 and ∑ _{r} α_{rc} ⩽ 1 for all c and ∑ _{c} α_{rc} ⩽ 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}, …, c_{k}, r_{k}) ∼ (α, β) be a closed walk.

(a) If r_{i} = r_{j} (vs. c_{i} = c_{j}) for some i ≠ j, then P is the juxtaposition of two nonempty closed walks P_{1} ∼ (α^{1}, β^{1}) and P_{2} ∼ (α^{2}, β^{2}) such that (α, β) = (α^{1}, β^{1})+(α^{2}, β^{2}) and ∑ _{c} α_{r_{i} c}^{1} , ∑ _{c} α_{r_{i} c}^{2} ⩾ 1 (vs. ∑ _{r} α_{rc_{i}}^{1} , ∑ _{r} α_{rc_{i}}^{2} ⩾ 1.)

(b) P is the juxtaposition of nonempty closed walks P_{j} ∼ (α^{j} , β^{j}) such that ∑ _{r} α_{rc}^{j} ⩽ 1 for all c, ∑ _{c} α_{rc}^{j} ⩽ 1 for all r and (α, β) = ∑ (α^{j} , β^{j}).

(c) There are cycles P_{j} ∼ (α^{j} , β^{j}) and a γ such that (α, β) = (γ, γ) + ∑ (α^{j} , β^{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}, …, c_{i}, r_{i}) and P_{2} = (c_{i+1} , r_{i+1} , …, 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}, …, c_{j−1} , r_{j−1} ) and P_{2} = (c_{j} , r_{j} , …, 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 ∼ (e_{q}, e_{q}) for some q ∈ I.

7 Schur multipliers on a cycle

We can realise a cycle of even length 2s, s ⩾ 2, in the following convenient way. Let Γ = ℤ/sℤ. Then the adjacency relation of integers modulo s turns Γ into the cycle (0, 1, …, s − 1) of length s. We double this cycle into the bipartite cycle (col0, row0, col1, row1, …, cols − 1 , rows − 1 ) on Γ ⨿ Γ, corresponding to the set of couples I = {(i, i), (i, i + 1) : i ∈ Γ} ⊆ Γ × Γ: I is the pattern

( \begin{matrix}& 0 & 1 & 2 & ··· & s−2 & s−1 \\ 0 & ∗ & ∗ & 0 & ... & 0 & 0 \\ 1 & 0 & ∗ & ∗ & ... & 0 & 0 \\ 2 & 0 & 0 & ∗ & ... & 0 & 0 \\ ⋮ & ... & ... & ... & ... & ... & ... \\ s−2 & 0 & 0 & 0 & ... & ∗ & ∗ \\ s−1 & ∗ & 0 & 0 & ... & 0 & ∗ \end{matrix} ) .

Γ is the group dual to G = Γ̂ = {z ∈ ℂ : z^{s} = 1}. We shall consider the space L_{Λ}^{p} (G) spanned by Λ = {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} ∑ _{z^{s} =1} |a + bz|^{p})^{1/p}.

Proposition 7.1. Let 0 < p ⩽ ∞, s ⩾ 2 and I = {(i, i), (i, i + 1) : i ∈ ℤ/sℤ}. Let ϵ ∈ 𝕋^{I} be a Schur multiplier by signs on S_{I}^{p}.

(a) The Schur multiplier ϵ has the same norm as the Schur multiplier ϵ̂ given by ϵ̂_{q} = 1 for q ≠ (s − 1 , 0) and ϵ̂_{s−1,0} = \overline{ϵ_{00}}ϵ_{01} … \overline{ϵ_{s−1 ,s−1}} ϵ_{s−1 ,0}.

(b) The Schur multiplier ϵ has the same norm as ϵ̌ given by ϵ̌_{ii} = 1 and ϵ̌_{i,i+1} = ϑ with ϑ any sth root of ϵ̂_{s−1 ,0} or its complex conjugate: without loss of generality, ϑ = e^{iα} with α ∈ [0, π/s].

(c) The norm of ϵ on S_{I}^{p} is bounded below by the norm of the relative Fourier multiplier μ : a+bz ↦ a + ϑbz on L_{Λ}^{p} (G); their complete norms are equal.

(d) The norm of ϵ on S_{I}^{1} and on S_{I}^{∞} is equal to the norm of μ on L_{Λ}^{1} (G) and on L_{Λ}^{∞} (G): this norm

is

\frac{cos(α/2 − π/2s)}{cosπ/2s} = \frac{max_{z^{s} =−1} |ϑ + z|}{|1 + e^{iπ/s} |} .

(e) The Schur multiplication operator M_{ϵ} is an isometry on S_{I}^{p} if and only if p/2 ∈ {1, 2, …, s − 1} or \overline{ϵ_{00}}ϵ_{01} … \overline{ϵ_{s−1 ,s−1}} ϵ_{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 ϵ and ϵ̄= (\overline{ϵ_{q}})_{q∈I} have the same norm on S_{I}^{p}.

(c) follows from Prop. 2.5.

(d). Let us compute f(β) = ∥1 + e^{iβ} z∥ _{L^{1} (G)} . As f(β) = f(β + 2π/s) = f(−β), we may suppose without loss of generality that β ∈ [0, π/s]. Then |β/2 + kπ/s| ⩽ π/2 if −⌊s/2⌋ ⩽ k ⩽ ⌈s/2⌉ − 1, so that

f(β) = \frac{1}{s} ∑_{k=−⌊s/2⌋}^{⌈s/2⌉−1} |1 + e^{iβ} e^{2ikπ/s} |

= \frac{2}{s} ∑_{k=−⌊s/2⌋}^{⌈s/2⌉−1} cos(β/2 + kπ/s)

= \frac{2}{s}ℜ( e^{iβ/2} \frac{e^{i⌈s/2⌉π/s} − e^{−i⌊s/2⌋/s}}{e^{iπ/s} − 1} )

= \frac{2}{s sin(π/2s)} · \{ \begin{matrix}cos(β/2 − π/2s) & if s is even \\ cos(β/2) & if s is odd. \end{matrix}

This shows in both cases that the norm of μ on L_{Λ}^{1} (G) is bounded below by cos(α/2 − π/2s)/ cos(π/2s). The complete norm of μ on L_{Λ}^{∞} (G) is equal to its norm and thus to the maximum of g(w) = ∥w + ϑz∥ _{L^{∞} (G)} /∥w + z∥ _{L^{∞} (G)} for w ∈ ℂ. Let w = re^{iβ} with r ⩾ 0 and β ∈ ℝ. Note that

∥w + z∥ _{L^{∞} (G)} = |r + e^{id(β,(2π/s)ℤ)} |

is a decreasing function of d(β, (2π/s)ℤ) and that

d(α − β, (2π/s)ℤ) < d(β, (2π/s)ℤ) ⇔ β ∈ ]α/2, π/s + α/2[ mod 2π/s.

As g(w) = g(wz) if z^{s} = 1, we may suppose without loss of generality that β ∈ ]α/2, π/s + α/2[. Therefore

g(w) = \{ \begin{matrix}| \frac{w + e^{iα}}{w + 1} | & if β ∈ ]α/2, π/s] \\ | \frac{w + e^{iα}}{w + e^{2iπ/s}} | & if β ∈ [π/s, π/s + α/2[. \end{matrix}

As g tends to 1 at infinity and g(w) = 1 if β ∈ {α/2, π/s + α/2}, the maximum principle shows that g attains its maximum with β = π/s. Finally,

g(re^{iπ/s} )^{2} = \frac{1 + 2r cos(π/s − α) + r^{2}}{1 + 2r cos(π/s) + r^{2}}

= 1 + \frac{cos(π/s − α) − cosπ/s}{cos(π/s) + (r + 1/r)/2} ⩽ g(e^{iπ/s} )^{2} = ( \frac{cos(π/2s − α/2)}{cosπ/2s} ) ^{2}.

(e). If p is not an even integer and ϑ^{s} ≠ 1, then μ is not an isometry on L_{Λ}^{p} (G): otherwise the functions z and ϑz would have the same distribution by the Plotkin-Rudin Equimeasurability Theorem (see [11, Th. 2]). If p ∈ {2, 4, …, 2s − 2}, then I contains no cycle of length 4, 6, …, p, so that by Prop. 6.6(c) every closed walk P ∼ (α, β) satisfies α = β. The function Φ_{I} (ϵ, a) in Eq. (9) is therefore constant in ϵ by Th. 6.5(a). If p ∈ {2s, 2s + 2, …}, the closed walk relation

(α, β) = ( ∑_{i∈Γ} e_{ii} , ∑_{i∈Γ} e_{i,i+1} ) + (p/2 − s)(e_{00}, e_{00})

satisfies n_{αβ} ⩾ 1 by Prop. 6.4. Then the coefficient of Φ_{I} (ϵ, a) in ā^{α} a^{β} equals

n_{αβ} \overline{ϵ_{00}}ϵ_{01} … \overline{ϵ_{s−1 ,s−1}} ϵ_{s−1 ,0}

and must equal the same quantity with ϵ replaced by 1 if ϵ defines an isometry on S_{I}^{p}.

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 ϵ̌ with α = π/s, and the complex unconditional constant is the maximum of the norm of ϵ̌ for α ∈ [0, π/s]. This yields

Corollary 7.3. Let 0 < p ⩽ ∞ and s ⩾ 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 ∈ {2, 4, …, 2s − 2}.

(b) The real and complex unconditional constants of I in the spaces S^{1} and S^{∞} equal secπ/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^{∞} (Section 5) and its generalisation to S^{p} for p not an even integer.

Theorem 8.1. Let I ⊆ R × C be nonempty and p ∈ (0, ∞] \ 2ℕ. 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 ϵ ∈ 𝕋^{I} there are ζ ∈ 𝕋^{C} and η ∈ 𝕋^{R} such that ϵ_{rc} = ζ(c)η(r) for all (r, c) ∈ I.

(f) For each ϵ ∈ {−1, 1}^{I} there are ζ ∈ {−1, 1}^{C} and η ∈ {−1, 1}^{R} such that ϵ_{rc} = ζ(c)η(r) for all (r, c) ∈ I.

(g) I is a set of V-interpolation of constant 1: for all φ ∈ ℓ_{I}^{∞}

inf{ ∥ ∑_{(r,c)∈R×C} φ̃_{rc} e_{c} ⊗ e_{r} ∥_{ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞}} : φ̃|_{I} = φ} = sup_{q∈I}|φ_{q}|.  (15)

(h) I is a V-Sidon set of constant 1: for all φ ∈ c_{0}(I)

inf{ ∥ ∑_{(r,c)∈R×C} φ̃_{rc} e_{c} ⊗ e_{r} ∥_{c_{0} (C) ⊗^{∧} c_{0} (R)} : φ̃|_{I} = φ} = sup_{q∈I}|φ_{q}|.  (16)

(i) For every tensor u = ∑ _{(r,c)∈I} a_{rc} e_{c} ⊗ e_{r} in ℓ_{C}^{1} ⊗^{∨} ℓ_{R}^{1} with support in I we have ∥u∥_{ℓ_{C}^{1} ⊗^{∨} ℓ_{R}^{1}} =

∑ _{(r,c)∈I} |a_{rc}|.

(j) (z_{c}z′_{r})_{(r,c)∈I} is a Sidon set of constant 1 in the dual of 𝕋^{C} × 𝕋^{R}, that is, a 1-unconditional basic sequence in C(𝕋^{C} × 𝕋^{R}): if (a_{rc}) is finitely supported,

sup_{(z,z′)∈𝕋^{C} ×𝕋^{R}} | ∑_{(r,c)∈I} a_{rc}z_{c}z′_{r}| = ∑_{(r,c)∈I} |a_{rc}| .

(k) For all R′ ⊆ R and C′ ⊆ C with k ⩾ 1 elements # [I ∩ R′× C′] ⩽ 2k − 1.

(l) I is an isometric interpolation set for Schur multipliers on S^{∞} : every φ ∈ ℓ_{I}^{∞} is the restriction of a Schur multiplier on S^{∞} with norm ∥M_{φ} ∥ = ∥φ∥_{ℓ_{I}^{∞}} .

Proof. (a) ⇒ (b) ⇒ (c) is trivial.

(c) ⇒ (d). Suppose that I contains a cycle (c_{0}, r_{0}, …, c_{s−1} , r_{s−1} ) with s ⩾ 2. Cor. 7.3(a) shows that I is not a real 1-unconditional basic sequence in S^{p}.

(d) ⇔ (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) ⇒ (e). Suppose first that I is a tree and index the vertices of its edges by words w ∈ W as described in the Terminology. Let us define η and ζ inductively. If r is the root of the tree, indexed by ∅, let η(r) = 1. Suppose that η and ζ 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 ζ(c) = ϵ(r, c)/η(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 η(r) = ϵ(r, c)/ζ(c). If I is a union of pairwise disjoint trees, we may define η and ζ on each tree separately. We may finally extend η to R and ζ to C in an arbitrary manner.

(d) ⇒ (f) may be proved as (d) ⇒ (e).

(f) ⇒ (c). If (f) holds, then every Schur multiplier by signs ϵ ∈ {−1, 1}^{I} is elementary in the sense that ϵ = ζ ⊗ η. The complete norm of M_{ϵ} on any S_{I}^{p} is therefore ∥ζ∥_{ℓ_{C}^{∞}} ∥η∥_{ℓ_{R}^{∞}} = 1.

(e) ⇒ (g). If (e) holds, every φ ∈ 𝕋^{I} ⊆ ℓ_{I}^{∞} may be extended to an elementary tensor ζ ⊗ η of norm 1. (g) follows because every element of ℓ_{I}^{∞} with norm 1 is the half sum of two elements of 𝕋^{I}: note that e^{it} cosu = (e^{i(t+u)} + e^{i(t−u)} )/2.

(g) ⇒ (h). It suffices to check Equality (16) for φ with support contained in a finite rectangle set

R′× C′. As ℓ_{C′}^{∞} ⊗^{∧} ℓ_{R′}^{∞} is a subspace of ℓ_{C}^{∞} ⊗^{∧} ℓ_{R}^{∞} , Eq. (15) yields Eq. (16).

(h) ⇔ (i) because they are dual statements.

(i) ⇔ (j). Use Equality (8).

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

(l) ⇒ (a). Taking sign sequences φ ∈ 𝕋^{I} in (l) shows that all relative Schur multipliers by signs on S_{I}^{∞} define isometries. Apply Prop. 4.3.

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

Let us now answer Question 1.3.

Corollary 8.3. Let I ⊆ R × C. The following are equivalent.

(a) For all φ ∈ c_{0}(I) one has ∥∑ _{(r,c)∈I} φ_{rc} e_{c} ⊗ e_{r} ∥_{c_{0} (C) ⊗^{∧} c_{0} (R)} = sup_{q∈I} |φ_{q}|.

(b) There are pairwise disjoint sets R_{j} ⊆ R and pairwise disjoint sets C_{j} ⊆ C such that R_{j} or C_{j} is a singleton for each j and I = ⋃ R_{j} ×C_{j}: I is the union of the column section ⋃ _{# R_{j} =1} R_{j} ×C_{j} with the disjoint row section ⋃ _{# R_{j} >1} R_{j} × C_{j}.

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

Proof. (a) ⇒ (b) follows from Prop. 3.1(a) ⇒ (d) and Th. 8.1(g) ⇒ (d).

(b) ⇔ (c). (b) holds if and only if (r, c), (r′, c), (r, c′) ∈ I ⇒ (r = r′ or c = c′) and therefore if and only if (c) holds.

(b) ⇒ (a). Suppose (b) and let φ ∈ c_{0}(I). Let α_{j} = sup_{(r,c)∈R_{j} ×C_{j}} |φ_{rc}|^{1/2} . If α_{j} = 0, let us define ϱ^{j} = 0 and γ^{j} = 0. Otherwise, if R_{j} is a singleton {r}, let us define ϱ^{j} = α_{j} e_{r} and γ^{j} by γ_{c}^{j} = φ_{rc}/α_{j} if c ∈ C_{j} and γ_{c}^{j} = 0 otherwise. Otherwise, C_{j} is a singleton {c} and we define γ^{j} = α_{j} e_{c} and ϱ^{j} by ϱ_{r}^{j} = φ_{rc}/α_{j} if r ∈ R_{j} and ϱ_{r}^{j} = 0 otherwise. Note that the γ^{j} have pairwise disjoint support and are null sequences, as well as the ϱ^{j}. Then

∑_{(r,c)∈I} φ_{rc} e_{c} ⊗ e_{r} = ∑_{j} γ^{j} ⊗ ϱ^{j} = average_{ϵ_{j} =±1}( ∑_{j} ϵ_{j} γ^{j} ) ⊗ ( ∑_{j} ϵ_{j} ϱ^{j} )

is an average of elementary tensors in c_{0}(C) ⊗^{∧} c_{0}(R) of norm sup_{q∈I} |φ_{q}|, so that this average is also bounded by this norm, which obviously is a lower bound.

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 ⊆ R × 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 ⊆ I there is an operator x ∈ S^{p}, whose support S contains F, such that ∥∑ ϵ_{q}x_{q}e_{q}∥_{p} does not depend on the complex choice of signs ϵ ∈ 𝕋^{S}.

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

(e) For every finite subset F ⊆ I there is an operator x ∈ S^{p} with real matrix coefficients, whose support S contains F, such that ∥∑ ϵ_{q}x_{q}e_{q}∥_{p} does not depend on the real choice of signs ϵ ∈ {−1, 1}^{S}.

(f) Every closed walk P ∼ (α, β) of length 2s ⩽ 2k in I satisfies α = β.

(g) I does not contain any cycle of length 2s ⩽ 2k as a subgraph.

(h) For each v, w ∈ V there is at most one path in I of length l ⩽ k that joins v to w.

Proof. (a) ⇒ (b) ⇒ (c), (b) ⇒ (d) ⇒ (e) are trivial.

(c) ⇒ (g). Suppose that I contains a cycle P ∼ (γ, δ) of length 2s ⩽ 2k: the corresponding set of couples is F = {q : γ_{q} + δ_{q} = 1}. Let x be as in (c) and let (α, β) = (γ, δ) + (k − s)(e_{q}, e_{q}) for some arbitrary q ∈ F. Then (α, β) ∈ 𝒲_{k}^{S}. Consider f(ϵ) = ∥∑ ϵ_{q}x_{q}e_{q}∥_{p}^{p} as a function on the group 𝕋^{S}.

Then the Fourier coefficient f̂(ϵ^{β−α} ) of f at the Steinhaus character ϵ^{β−α} is, by Th. 6.5(a),

∑ \{n_{εζ} \overline{x}^{ε}x^{ζ} : (ε, ζ) ∈ 𝒲_{k}^{S} and ζ − ε = β − α\}

= \overline{x}^{γ}x^{δ} ∑ \{n_{εζ} \overline{x}^{ε−γ} x^{ζ−δ} : (ε, ζ) ∈ 𝒲_{k}^{S} and ζ − δ = ε − γ\}.

(Note that β − α = δ − γ.) As this last sum has only positive terms and contains at least the term corresponding to (α, β), f cannot be constant.

(e) ⇒ (g). Let P ∼ (γ, δ), F = {q : γ_{q} + δ_{q} = 1} and (α, β) be as in the proof of the implication (c) ⇒ (h). Let x be as in (e). Consider f(ϵ) = ∥∑ ϵ_{q}x_{q}e_{q}∥_{p}^{p} as a function on the group {−1, 1}^{S}.

Then the Fourier coefficient f̂(ϵ^{β−α} ) of f at the Walsh character ϵ^{β−α} is, by Th. 6.5(a),

∑ {n_{εζ}x^{ε+ζ} : (ε, ζ) ∈ 𝒲_{k}^{S} and ζ − ε ≡ β − α (mod 2)}

= x^{γ+δ} ∑ {n_{εζ}x^{ε+ζ−γ−δ} : (ε, ζ) ∈ 𝒲_{k}^{S} and ζ − ε ≡ δ − γ (mod 2)}.

As this last sum has only positive terms and contains at least the term corresponding to (α, β), f cannot be constant.

(f) ⇔ (g). Apply Prop. 6.6(c).

(g) ⇔ (h). If I contains a cycle (v_{0}, …, v_{2s−1} ), then I contains two distinct paths (v_{0}, …, v_{s}), (v_{0}, v_{2s−1} , …, v_{s}) of length s from v_{0} to v_{s}. If I contains two distinct paths (v_{0}, …, v_{l}), (v′_{0}, …, v′_{l′}) with v_{0} = v′_{0}, v_{l} = v′_{l′} and l, l′ ⩽ k, let a be minimal such that v_{a} ≠ v′_{a}, let b ⩾ a be minimal such that v_{b} ∈ {v′_{a}, …, v′_{l′}} and let d ⩾ a be minimal such that v′_{d} = v_{b}. Then (v_{a−1} , …, v_{b}, v′_{d−1} , …, v′_{a}) is a cycle in I of length 2s ⩽ 2k.

(f) ⇒ (a) holds by Theorem 6.5(b): If each (α, β) ∈ 𝒲_{k}^{I} satisfies α = β, then Eq. (14) shows that Ψ_{I} (ϵ, z) as defined in Eq. (13) is constant in ϵ.

Remark 9.2. The equivalence (b) ⇔ (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}, …, c_{s}, r_{s}) ∼ (α, β) in ℕ × ℕ is considered as the polygonal closed curve γ in ℂ with sides parallel to the coordinate axes whose successive vertices are r_{1} + ic_{1}, r_{1} + ic_{2}, r_{2} + ic_{2}, …, r_{s−1} + ic_{s}, r_{s} + ic_{s}, r_{s} + ic_{1} and again r_{1} + ic_{1}. Then α = β if and only if the index with respect to γ of every point not on γ is zero, if and only if γ 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 × 2 matrix x. Then trx^{∗}x = ∑ |x_{q}|^{2} and detx^{∗}x = |x_{00}x_{11} − x_{01}x_{10}|^{2}. This shows that

if ℜ(\overline{x_{00}x_{11}} x_{01}x_{10}) = 0, e.g. x = ( \begin{matrix}1 & 1 \\ 1 & i \end{matrix}) , then the singular values of x do not depend on the real

sign of the matrix coefficients of x, whereas (col0, row0, col1, row1) is a cycle of length 4.

Remark 9.5. Theorem 9.1(h) ⇒ (a) is the isometric counterpart to [9, Th. 3.1], which shows in particular that I is an unconditional basic sequence in 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 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^{∞} cannot even be embedded in a space with unconditional basis. If 1 < p < ∞, then S^{p} has the unconditional finite dimensional decomposition

⨁_{n∈ℕ} S_{{(r,c):r⩽n,c=n}}^{p} ⊕ S_{{(r,c):r=n+1,c⩽n}}^{p}

because the triangular projection associated to the idempotent Schur multiplier (χ_{r⩽c}) is bounded on S^{p}.

Definition 10.1. Let X be a separable Banach space and 𝕊 = 𝕋 (vs. 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∥ → 0 for every x ∈ X. An approximating sequence of commuting projections is a finite-dimensional decomposition.

• ([18].) The difference sequence (ΔT_{k}) of (T_{k}) is given by ΔT_{1} = T_{1} and ΔT_{k} = T_{k} − T_{k−1} for k ⩾ 2. X has the unconditional approximation property (uap) if there is an approximating sequence (T_{k}) such that for some constant D

∥∑_{k=1}^{n} ϵ_{k}ΔT_{k} ∥ ⩽ D for all n and ϵ_{k} ∈ 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 δ > 0, X has an approximating sequence with complex (vs. real) unconditional constant 1 + δ. 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_{ϵ∈𝕊}∥T_{k} + ϵ(Id − T_{k})∥ −→ 1.  (17)

X has (muap) if and only if, for every given δ > 0, X is isometric to a 1-complemented subspace of a space with a (1+δ)-unconditional finite-dimensional decomposition [6, Cor. IV.4]. If X has (muap), then, for any given δ > 0, X is isometric to a subspace of a space with a (1 + δ)-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 ⊆ R × C is such that all columns I ∩ R × {c} (vs. all rows I ∩ {r} × C) are finite, then S_{I}^{p} admits a 1-unconditional finite-dimensional decomposition in the corresponding finitely supported idempotent Schur multipliers χ_{I∩R×{c}} (vs. χ_{I∩{r}×C} .)

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

Theorem 10.3. Let 1 ⩽ p ⩽ ∞. Let R_{r} ⊆ R, r ∈ ℕ, be pairwise disjoint and finite. Let C_{c} ⊆ C, c ∈ ℕ, be pairwise disjoint and finite. Let J ⊆ ℕ × ℕ and I = ⋃ _{(r,c)∈J} R_{r} × C_{c}. Then the sequence of Schur multipliers (χ_{R_{r} ×C_{c}} )_{(r,c)∈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 ⩽ p ⩽ ∞ and I ⊆ R × C. Let (T_{n}) be an approximating sequence on S_{I}^{p}. Then there is a sequence of Schur multipliers (φ_{n}) such that (M_{φ_{n}} ) is an approximating sequence on S_{I}^{p} and such that if (T_{n}) satisfies (17), then so does (M_{φ_{n}} ).

Proof. Let δ_{n} > 0 be such that δ_{n} → 0. As T_{n} has finite rank, there is a finite R_{n} × C_{n} ⊆ R × C such that the projection P_{R_{n} ×C_{n}} of S^{p} onto S_{R_{n} ×C_{n}}^{p} defined by the Schur multiplier χ_{C_{n}} ⊗ χ_{R_{n}} satisfies ∥P_{R_{n} ×C_{n}} T_{n} − T_{n}∥ < δ_{n}. Let φ_{n} be the Schur multiplier associated to [T_{n}]_{R_{n} ×C_{n}} . With the notation of Eq. (5),

M_{φ_{n}} (x) − x = ∫_{𝕋^{R}} dη ∫_{𝕋^{C}} dζ M_{ζ^{∗} ⊗η^{∗}} (P_{R_{n} ×C_{n}} T_{n} − Id)(M_{ζ⊗η} x).

As P_{R_{n} ×C_{n}} T_{n} tends to the identity uniformly on compact sets, this shows that M_{φ_{n}} is an approximating sequence. As

M_{φ_{n}} + ϵ(Id − M_{φ_{n}} ) = [P_{R_{n} ×C_{n}} T_{n} + ϵ(Id − P_{R_{n} ×C_{n}} T_{n})] ,

the norm of this operator is at most ∥T_{n} + ϵ(Id − T_{n})∥ + 2δ_{n}.

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

Corollary 10.5. Let 1 ⩽ p ⩽ ∞ and I ⊆ R × C.

• If S_{I}^{p} has (muap), then some sequence of Schur multipliers realises it.

• Let J ⊆ I. If S_{I}^{p} has (muap), then so does S_{J}^{p}.

• If S_{I}^{∞} has (muap), then so does S_{I}^{p}.

Let us define the following asymptotic properties.

Definition 10.6. Let 1 ⩽ p ⩽ ∞, I ⊆ R × C and 𝕊 = 𝕋 (vs. 𝕊 = {−1, 1}.)

• S_{I}^{p} is asymptotically unconditional if for every x ∈ 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_{ϵ∈𝕊}∥x + ϵy_{n}∥_{p} − min_{ϵ∈𝕊}∥x + ϵy_{n}∥_{p} −→ 0.

• I enjoys the property (𝒰 ) of block unconditionality in S^{p} if for each δ > 0 and finite F ⊆ I, there is a finite G ⊆ I such that

∀ x ∈ B_{S_{F}^{p}} ∀ y ∈ B_{S_{I\G}^{p}} max_{ϵ∈𝕊}∥x + ϵy∥_{p} − min_{ϵ∈𝕊}∥x + ϵy∥_{p} < δ.

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

Theorem 10.7. Let 1 ⩽ p ⩽ ∞, I ⊆ R × C and 𝕊 = 𝕋 (vs. 𝕊 = {−1, 1}.) Consider the following properties.

(a) S_{I}^{p} is asymptotically unconditional.

(b) I enjoys (𝒰 ) in S^{p}.

(c) S_{I}^{p} has (muap).

Then (c) ⇒ (a) ⇔ (b). If 1 < p < ∞, then (b) ⇔ (c). If p = 1, S_{I}^{1} has (muap) if and only if S_{I}^{1} has (uap) and I enjoys (𝒰 ) in S^{1}.

The case p = ∞ is extreme in the sense that the following properties are equivalent for S_{I}^{∞} : 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}^{∞} is isomorphic to ℓ_{I}^{2}.

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

Definition 10.8. Let I ⊆ R × C and k ⩾ 1.

• I enjoys property 𝒥_{k} if for every path P = (c_{0}, r_{0}, …, c_{j} , r_{j}) of odd length 2j + 1 ⩽ k in I there is a finite set R′× C′ such that P cannot be completed with edges in I \ R′× C′ to a cycle of length 2s ∈ {4j + 2, …, 2k}.

• The asymptotic distance d_{∞} (r, c) of r ∈ R and c ∈ C in I is the supremum, over all finite rectangle sets R′× C′, of the distance from r to c in I \ R′× C′.

The asymptotic distance takes its values in {1, 3, 5, …, ∞}. Note that 𝒥_{1} is true and that 𝒥_{k} ⇒ 𝒥_{k−1} . This implication is strict: let R, C be two copies of ℕ and, given j ⩾ 1, consider the union I_{j} of all paths (col0, rownj + 1 , colnj + 1 , …, rownj + j , colnj + j , row0) of length 2j + 1. Then I_{j} contains no cycle of length 2s ∈ {4, …, 4j} and therefore enjoys 𝒥_{2j} , but fails 𝒥_{2j+1} ; I_{j} ∪ {(row0, col0)} contains no cycle of length 2s ∈ {4, …, 2j} and thus enjoys 𝒥_{j}, but fails 𝒥_{j+1} . In particular, the properties 𝒥_{k}, k ⩾ 2, are not stable under union with a singleton.

Let us now explicit the relationship between 𝒥_{k} and d_{∞} .

Proposition 10.9. Let I ⊆ R × C and k ⩾ 1.

(a) I enjoys 𝒥_{k} if and only if any two vertices r ∈ R and c ∈ C at distance 2j + 1 ⩽ k satisfy d_{∞} (r, c) ⩾ 2k − 2j + 1.

(b) If d_{∞} (r, c) ⩾ 2k + 1 for all (r, c) ∈ R × C, then I enjoys 𝒥_{k}.

(c) If d_{∞} (r, c) ⩽ k for some (r, c) ∈ R × C, then I fails 𝒥_{k}.

(d) I enjoys 𝒥_{k} for every k if and only if d_{∞} (r, c) = ∞ for every (r, c) ∈ R × C.

Proof. (a) is but a reformulation of the definition of𝒥_{k} and implies (b).

(d) is a consequence of (b) and (c).

(c). If d_{∞} (r, c) ⩽ k, then there is 0 ⩽ j ⩽ (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}, …, r_{j} , c_{j} , r) that can be completed with edges outside any given finite set to a cycle of length 4j + 2 ⩽ 2k.

Theorem 10.10. Let I ⊆ R × C and 1 ⩽ p ⩽ ∞. If p is an even integer, then S_{I}^{p} has complex or real (muap) if and only if I enjoys 𝒥_{p/2}. If p = ∞ or if p is not an even integer, then S_{I}^{p} has real (muap) only if I enjoys 𝒥_{k} for every k.

Proof. Suppose that I enjoys (𝒰 ) in S^{p} and fails 𝒥_{k}. Then, for some s ⩽ k, I contains a sequence of cycles (c_{0}, r_{0}, …, c_{j−1} , r_{j−1} , c_{j}^{n}, r_{j}^{n}, …, c_{s−1}^{n} , r_{s−1}^{n} ) with the property that ∥x−y∥_{p} ⩽ (1+1/n)∥x+y∥_{p} for all x with support in {(r_{0}, c_{0}), (r_{0}, c_{1}), …, (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}), …, (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)} ⊆ ℤ/sℤ × ℤ/sℤ given by ϵ_{rc} = 1 if r, c ∈ \{0, …, j − 1\} and ϵ_{rc} = −1 otherwise actually is an isometry on S_{I}^{p}. As \overline{ϵ_{00}}ϵ_{01} … \overline{ϵ_{s−1 s−1}} ϵ_{s−1 0} = (−1)^{2s−2j+1} = −1, this implies by Prop. 7.1(e) that p/2 ∈ {1, 2, …, s − 1}.

Suppose that I enjoys 𝒥_{k}. We claim that for every finite F ⊆ I there is a finite G ⊆ I such that every closed walk P ∼ (α, β) of length 2k in I satisfies ∑ _{q∈I\G} β_{q} − α_{q} = 0. This signifies that given a closed walk (v_{0}, …, v_{2k−1} ) and 0 = a_{0} < b_{0} < ⋯< a_{m} < b_{m} < a_{m+1} = 2k such that v_{a_{i}} , …, v_{b_{i} −1} ∈ I \ G and v_{b_{i}}, …, v_{a_{i+1} −1} ∈ F,

{i ∈ {0, …, m} : a_{i}, b_{i} even} = {i ∈ {0, …, m} : a_{i}, b_{i} odd}.

Suppose that this is not true: then there is an s ⩽ k, there are 0 = a_{0} < b_{0} < ⋯< a_{m} < b_{m} < 2s and there are cycles (v_{a_{0}}^{n} , …, v_{b_{0} −1}^{n} , v_{b_{0}}, …, v_{a_{1} −1} , …, v_{a_{m}}^{n} , …, v_{b_{m} −1}^{n} , v_{b_{m}} , …, v_{2s−1} ) such that the (v_{i}^{n})_{n⩾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} ⩾ s − 1, consider the path P = (v_{b_{0}}, …, v_{a_{0} −1} , v_{a_{0}}^{0} , …, v_{b_{m} −1}^{0} , v_{b_{m}} , …, v_{2s−1} ) of odd length 2s − 1 − (b_{0} − a_{0}); if b_{0} − a_{0} ⩽ s − 1, consider the path P = (v_{2s−1} , v_{a_{0}}^{0} , …, 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} , …, 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 𝒥_{s}.

The claim shows that I enjoys (𝒰 ) in S^{p} for p = 2k. In fact, if ϵ̃∈ 𝕋^{F ∪(I\G)} is defined by ϵ̃_{q} = 1 for q ∈ F and ϵ̃_{q} = ϵ ∈ 𝕋 for q ∈ I \ G, then, with the notation of Th. 6.5,

Φ_{F ∪(I\G)} (ϵ̃, a) = ∑_{(α,β)∈𝒲_{k}^{F ∪(I\G)}} n_{αβ} ϵ^{∑ _{q∈I\G} β_{q}−α_{q}} \overline{a}^{α} a^{β}

does not depend on ϵ, so that ∥x + ϵy∥_{2k} = ∥x + y∥_{2k} if x ∈ S_{F}^{2k} and y ∈ S_{I\G}^{2k} , and S_{I}^{2k} has complex (muap) by Th. 10.7(b) ⇒ (c).

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 ℓ_{∞} ⊗^{∧} ℓ_{∞} are lacunary sets in a locally compact abelian group.

Let Γ be a discrete abelian group and Λ ⊆ Γ. Let us say that Λ is n-independent if every element of Γ admits at most one representation as the sum of n terms in Λ, up to a permutation. For example, the geometric sequence {j^{k}}_{k⩾0} with j ∈ {2, 3, …} is n-independent in ℤ if and only if j ⩾ n [14, § 3]. If Λ is n-independent for all n, then Λ is independent. Let

Z_{n} = {ζ ∈ ℤ^{Λ} : ∑_{γ∈Λ} ζ_{γ} = 0 and ∑_{γ∈Λ} |ζ_{γ}| ⩽ 2n]

and Z = ⋃ Z_{n}. Then Λ is n-independent if and only if, for every ζ ∈ Z_{n},

∑_{γ∈Λ} ζ_{γ}γ = 0 ⇒ ζ = 0

and Λ is independent if and only if this holds for every ζ ∈ Z.

Let us say that Λ is n-independent modulo 2 if in every representation of an element of Γ as the sum of n terms in Λ, each element of Λ appears the same number of times modulo 2. In other words, for every ζ ∈ Z_{n},

∑_{γ∈Λ} ζ_{γ}γ = 0 ⟹ ∀ γ ∈ Λ ζ_{γ} = 0 (mod 2) ;

Λ is independent modulo 2 if this holds for every ζ ∈ Z. If Γ contains no element of order 2, then one may always suppose that at least one coefficient ζ_{γ} of a nontrivial relation ∑ ζ_{γ}γ = 0 is odd, so that these two latter notions “modulo 2” coincide with the two former ones.

Let G = Γ̂, so that Γ is the group of characters on G. Then the computation presented in [14, Prop. 2.5(ii)] for the case Γ = ℤ shows that Λ is a complex (vs. real) 1-unconditional basic sequence in L^{p}(G) with p ∈ 2ℕ^{∗} if and only if Λ is p/2-independent (vs. modulo 2). Furthermore Λ is a complex (vs. real) 1-unconditional basic sequence in L^{p}(G) with p ∈ (0, ∞]\2ℕ^{∗} if and only if Λ is independent (vs. modulo 2). If Γ 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 Γ be a discrete abelian group and R, C be countable subsets of Γ. To every Λ ⊆ R + C associate I_{Λ} = {(r, c) ∈ R × C : r + c ∈ Λ}. Let G = Γ̂.

(a) If Λ is a complex 1-unconditional basic sequence in L^{4}(G), then I_{Λ} is a 1-unconditional basic sequence in S^{4}.

(b) Suppose that each element of Γ admits at most one representation as the sum of an element of R with an element of C. Then every I ⊆ R×C has the form I = I_{Λ} with Λ = {r + c : (r, c) ∈ I} . If Λ is a real 1-unconditional basic sequence in L^{p}(G), then I_{Λ} is a 1-unconditional basic sequence in S^{p}.

(c) Let p = 2k be a positive even integer. Suppose that R ∩ C = ∅ and R ∪ C is k-independent modulo 2. I_{Λ} is a 1-unconditional basic sequence in S^{p} if and only if Λ 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_{Λ} . Then r + c, r′+ c′, r + c′ and r′+ c are in Λ while (r + c) + (r′+ c′) = (r + c′) + (r′+ c): if Λ is 2-independent, then r + c ∈ {r + c′, r′+ c}, so that c = c′ or r = r′ and P is not a cycle.

(b). For each γ ∈ Λ, let q_{γ} = (r_{γ}, c_{γ}) be the unique element of I such that r_{γ} + c_{γ} = γ. If Λ 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 φ ∈ {−1, 1}^{I_{Λ}} , so that φ_{q_{γ}} ∈ {−1, 1} for all γ ∈ Λ. Then, as in Eq. (6),

∥ ∑_{q∈I_{Λ}} a_{q}e_{q}∥_{S_{I_{Λ}}^{p} (S^{p})} = ∥ ∑_{(r,c)∈I_{Λ}} r(g)c(g)a_{rc} e_{rc} ∥_{S_{I_{Λ}}^{p} (S^{p})}

= ∥∑_{γ∈Λ} a_{q_{γ}} e_{q_{γ}} γ(g)∥_{S_{I_{Λ}}^{p} (S^{p})} = ∥∑_{γ∈Λ} a_{q_{γ}} e_{q_{γ}} γ∥_{L_{Λ}^{p} (G,S^{p} (S^{p}))}

so that as in Eq. (2), by complete real 1-unconditionality of Λ in L^{p}(G),

∥ ∑_{q∈I_{Λ}} φ_{q}a_{q}e_{q}∥_{S_{I_{Λ}}^{p} (S^{p})} = ∥∑_{γ∈Λ} φ_{q_{γ}} a_{q_{γ}} e_{q_{γ}} γ∥_{L_{Λ}^{p} (G,S^{p} (S^{p}))} = ∥ ∑_{q∈I_{Λ}} a_{q}e_{q}∥_{S_{I_{Λ}}^{p} (S^{p})} .

(c). Each element of Γ 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 Λ is not a real 1-unconditional basic sequence in L^{p}(G) and let ζ ∈ Z_{k} such that ∑ _{γ∈Λ} ζ_{γ}γ = 0 and J = {(r, c) ∈ I_{Λ} : ζ_{r+c} ≠ 0 (mod 2)} is nonempty; J has at most 2k elements. Let P = (v_{1}, …, v_{j}) be a path in J of maximal length. Then ζ_{v_{j−1} +v_{j}} is odd and ∑ {ζ_{v_{j} +v} : v_{j} + v ∈ Λ} is even because it is the coefficient of v_{j} in the relation ∑ _{γ∈Λ} ζ_{γ}γ = 0 and R ∪ C is k-independent modulo 2. There is therefore v_{j+1} distinct from v_{j−1} such that ζ_{v_{j} +v_{j+1}} is odd. As j is maximal and R ∩ C = ∅, v_{j+1} = v_{j+1−2i} for some 2 ⩽ i ⩽ k, so that (v_{j+1−2i} , …, v_{j}) is a cycle of length 2i in J: I_{Λ} is not a 1-unconditional basic sequence in S^{p}.

Let R and C be any countable sets. Consider G = {−1, 1}^{C} ×{−1, 1}^{R}. If we denote by ((ϵ_{c})_{c∈C} , (ϵ′_{r})_{r∈R} ) a generic point in G, then the set of Rademacher functions {ϵ_{c}}_{c∈C} ∪ {ϵ′_{r}}_{r∈R} is a real 1-unconditional basic sequence in C(G), so that it is independent modulo 2 in Ĝ. Similarly, the set of Steinhaus functions {z_{c}}_{c∈C} ∪ {z′_{r}}_{r∈R} is independent in the dual of 𝕋^{C} × 𝕋^{R}. This yields:

Corollary 11.2. Let I ⊆ R × C and p ∈ (0, ∞]. The following are equivalent:

• I is a 1-unconditional basic sequence in S^{p}.

• {ϵ_{c}ϵ′_{r} : (r, c) ∈ I} is a real 1-unconditional basic sequence in L^{p}(G).

• {z_{c}z′_{r} : (r, c) ∈ I} is a 1-unconditional basic sequence in L^{p}(𝕋^{C} × 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 σ(p) set) if and only if {ϵ_{c}ϵ′_{r} : (r, c) ∈ I} is a completely unconditional basic sequence in L^{p}(G) (a Λ(p)_{cb} set in Ĝ, see [8] and [21, § 8.1],) if and only if {z_{c}z′_{r} : (r, c) ∈ I} is a completely unconditional basic sequence in L^{p}(𝕋^{C} × 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 ℕ × ℕ. We have in our context the rather disappointing

Proposition 11.4. Let Λ ⊆ ℕ ⊆ ℤ and I = {(r, c) ∈ ℕ × ℕ : r + c ∈ Λ}.

(a) I is a 1-unconditional basic sequence in S^{4} if and only if {z^{λ}}_{λ∈Λ} is a 1-unconditional basic sequence in L^{4}(𝕋).

(b) If Λ contains three elements λ < μ < ν such that λ + μ ⩾ ν, then I is not a 1-unconditional basic sequence in S^{p} if p ∈ (0, ∞] \ {2, 4}.

(c) If Λ = {λ_{k}} with λ_{k+1} > 2λ_{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 = ℕ. Conversely, if Λ contains a solution to λ + μ = λ′ + μ′ with λ < λ′ ⩽ μ′ < μ, then I contains the cycle (col0, rowλ, colλ′ − λ, rowμ′).

(b). Consider the cycle (col0, rowλ, colν − λ, rowμ − ν + λ, colν − μ, rowμ).

(c). In fact, I is a forest. Let P = (c_{1}, r_{1}, …, 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}, …, r_{k} + c_{k}, r_{k} + c_{1}}. Then r_{1}+c_{1} ⩽ r_{1}+c_{2} and r_{2}+c_{2} ⩽ 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} ∈ Λ, so that 2(r_{1} + c_{1} + r_{2} + c_{2}) < 2(r_{1} + c_{2}) and c_{1} + r_{2} < 0.

Remark 11.5. Further computations yield the following result. If {z^{λ}}_{λ∈Λ} is a 1-unconditional basic sequence in L^{6}(𝕋) and if {λ < μ < ν} ⊆ Λ ⇒ λ + μ < ν, 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 ⩽ n ⩽ m, I ⊆ R × C with # C = n and # R = m, and e = # I.

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

n ⩾ 1 + ( \frac{e}{m} − 1) + ( \frac{e}{m} − 1) ( \frac{e}{n} − 1) ,

that is e^{2} − me − mn(n − 1) ⩽ 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 ⩾ 1 + ( \frac{e}{m} − 1) + ( \frac{e}{m} − 1) ( \frac{e}{n} − 1) + ( \frac{e}{m} − 1)^{2}( \frac{e}{n} − 1) ,

that is e^{3} − (m + n)e^{2} + 2mne − m^{2}n^{2} ⩽ 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 ⩾ 1 an integer, then

n ⩾ ∑_{i=0}^{k} ( \frac{e}{m} − 1)^{⌈ \frac{i}{2} ⌉}( \frac{e}{n} − 1)^{⌊ \frac{i}{2} ⌋}.  (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 ⩾ 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].

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 ∈ {4, 6, 10, 14} [29, Lemma 1.7.1]; for 2k ∈ {6, 10, 14}, it follows from [29, Lemma 1.5.4] that

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

Remark 11.7. Let I ⊆ R × C with # C = # R = n. Inequality (18) shows that if I is a 1-uncon-ditional basic sequence in S^{2k}, then # I ⩽ n^{1+1/k} + (s − 1)n/s. If p ∉ {4, 6, 10}, the existence of 1-unconditional basic sequences in S^{2k} such that # I ≽ 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.

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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.