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MULTIPLIERS AND LACUNARY SETS IN NON-AMENABLE GROUPS

arXiv:math/9212207v1 [math.FA] 9 Dec 1992MULTIPLIERS AND LACUNARY SETS IN NON-AMENABLE GROUPSby Gilles Pisier*§ 0. Introduction.Let G be a discrete group.Let λ : G →B(ℓ2(G), ℓ2(G)) be the left regular representation.

A function ϕ : G →|Cis called a completely bounded multiplier (= Herz-Schur multiplier) if the transformationdefined on the linear span K(G) of {λ(x), x ∈G} byXx∈Gf(x)λ(x) →Xx∈Gf(x)ϕ(x)λ(x)is completely bounded (in short c.b.) on the C∗-algebra C∗λ(G) which is generated by λ(C∗λ(G) is the closure of K(G) in B(ℓ2(G), ℓ2(G)).

)One of our main results (stated below as Theorem 0.1) gives a simple characterizationof the functions ϕ such that εϕ is a c.b. multiplier on C∗λ(G) for any bounded function ε, orequivalently for any choice of signs ε(x) = ±1.

We wish to consider also the case when thisholds for “almost all” choices of signs. To make this precise, equip {−1, 1}G with the usualuniform probability measure.

We will say that εϕ is a c.b. multiplier of C∗λ(G) for almostall choice of signs ε if there is a measurable subset Ω⊂{−1, 1}G of full measure (note thatΩdepends only on countably many coordinates) such that for any ε in Ωεϕ is a c.b.multiplier of C∗λ(G).

(Note that ϕ is necessarily countably supported when this holds, so themeasurability issues are irrelevant. )Theorem 0.1.

The following properties of a function ϕ : G →|C are equivalent(i) For all bounded functions ε : G →|C the pointwise product εϕ is a c.b. multiplier.

(ii) For almost all choices of signs ε ∈{−1, 1}G, the product εϕ is a c.b. multiplier.

(iii) There is a constant C and a partition of G × G say G × G = Γ1 ∪Γ2 such thatsups∈GXt∈G|ϕ(st)|21{(s,t)∈Γ1} ≤C2 and supt∈GXs∈G|ϕ(st)|21{(s,t)∈Γ2} ≤C2. * Supported in part by N.S.F.

grant DMS 9003550

(iv) There is a constant C such that for all finite subsets E, F ⊂G with |E| = |F| = Nwe haveX(s,t)∈E×F|ϕ(st)|2 ≤C2N. (v) There is a constant C such that for any Hilbert space H and for any finitelysupported function a : G →B(H) we haveXx∈Gϕ(x)λ(x) ⊗a(x)B(ℓ2(G,H))≤C maxXa(x)∗a(x)1/2 ,Xa(x)a(x)∗1/2.Note :The properties (iii) and (iv) could have been stated equivalently with thefunction (s, t) →ϕ(st−1) (which would have been perhaps more natural)or (s, t) →ϕ(s−1t)instead of (s, t) →ϕ(st).

We chose the simplest notation.This theorem is proved in section 2 below.Remark : The papers [W] and [B3] show that amenable groups are characterized by theproperty that all multipliers ϕ satisfying (iii) in Theorem 0.1 are necessarily in ℓ2(G). Hencethe preceding statement is of interest only in the non-amenable case.

Moreover, on the freegroup with finitely many generators, the radial functions which satisfy the above property(iii) are characterized in [W].The equivalence of (iii) and (iv) is already known. It was proved by Varopoulos [V1]in his study of the projective tensor product ℓ∞ˆ⊗ℓ∞and the Schur multipliers of B(ℓ2, ℓ2).Let (es) (resp.

(et)) denote the canonical basis of ℓ∞(S) (resp. ℓ∞(T)).

We will denote by˜V (S, T) the set of functions ψ : S × T →|C such thatsupE⊂S,F ⊂T|E|<∞,|F |<∞XS∈Et∈Fψ(s, t)es ⊗etℓ∞(S) ˆ⊗ℓ∞(T )< ∞.More precisely, Varopoulos provedTheorem 0.2. ([V1]) Let S, T be arbitrary sets.

The following properties of a functionψ : S × T →|C are equivalent. (i) For all bounded functions ε : S × T →|C the pointwise product εψ is in ˜V (S, T).

(ii) For almost all choices of signs ε in {−1, 1}S×T the pointwise product εψ is in˜V (S, T).2

(iii) There is a constant C and a partition S × T = Γ1 ∪Γ2 such thatsups∈SXt∈T(s,t)∈Γ1|ψ(s, t)|2 ≤C2 and supt∈TXs∈S(s,t)∈Γ2|ψ(s, t)|2 ≤C2. (iii)’ There is a decomposition ψ = ψ1 + ψ2 withsups∈SXt∈T|ψ1(s, t)|2 < ∞and supt∈TXs∈S|ψ2(s, t)|2 ≤C.

(iv) There is a constant C such that for all finite subsets E ⊂S, F ⊂T with|E| = |F| = N, we haveX(s,t)∈E×F|ψ(s, t)|2 ≤C2N.The deepest implication in Theorem 0.2 is (ii) ⇒(iii). The equivalence of (iii) and (iii)’is obvious and (iii)’ ⇒(i) is rather easy (by duality, it follows from Khintchine’s inequality).The equivalence (iii) ⇔(iv) is a remarkable fact of independent interest.

The decompositionsof the form (iii) are related to some early work of Littlewood and the matrices admitting thedecomposition (iii) are often called Littlewood tensors, following Varopoulos’s terminology.We note in passing that (ii) ⇒(iii) (and in fact a slightly stronger result) can be obtainedas an application of Slepian’s comparison principle for Gaussian processes in the style of S.Chevet (see [C] th´eor`eme 3.2). However, we do not see how to exploit this approach in ourmore general context.We will prove below a result which contains Theorem 0.2 as a particular case and impliesTheorem 0.1 in the group case.

Roughly our result gives a necessary condition (analogousto the above (iii)) for a random series P∞n=1 εnψn with random signs εn = ±1 and arbitrarycoefficients ψn in ℓ∞ˆ⊗ℓ∞to define a.s. an element of ℓ∞ˆ⊗ℓ∞. The implication (ii) ⇒(iii)in Theorem 0.2 corresponds to the particular case when ψn is of the form ψn = αn ein ⊗ejnwhere αn ∈|C and n →(in, jn) is a bijection of IN onto IN × IN.Our necessary condition can be stated as follows : there is a sequence of scalars αmwith Pm≥0 |αm| < ∞and scalar coefficientsamn (i), bm(j), cm(i), dmn (j)3

such thatψn(i, j) =Xm≥0αm[amn (i)bm(j) + cm(i)dmn (j)]and such that for all msupi(Xn|amn (i)|2)1/2 ≤1,supj|bm(j)| ≤1supi|cm(i)| ≤1,supj(Xn|dmn (j)|2)1/2 ≤1.In other words, the condition expresses that the sequence (ψn) can be written (up to amultiplicative norming constant) as an element of the closed convex hull of special sequencesof the formψn(i, j) = an(i)b(j) + c(i)dn(j)withXn|an(i)|2 ≤1, |b(j)| ≤1, |c(i)| ≤1,Xn|dn(j)|2 ≤1for all i and j.This will be stated below (cf. Theorems 2.1 and 2.2) in the more precise (and concise)language of tensor products.To emphasize the content of Theorem 0.1, we now state an application in terms ofSchur multipliers.

For any sets S, T a function ψ : S × T →|C is called a Schur multiplierof B(ℓ2(S), ℓ2(T)) if for any u ∈B(ℓ2(S), ℓ2(T)) with associated matrix (u(s, t)) the matrix(ψ(s, t)u(s, t)) is the matrix of an element of B(ℓ2(S), ℓ2(T)). It is known that the set of allSchur multipliers ψ : S × T →|C coincides with the space ˜V (S, T).

This essentially goesback to Grothendieck [G]. We give more background on Schur multipliers in section 1.

Thenext statement is an application of Theorem 0.1 (and the easier implication (iii) ⇒(i) inTheorem 0.2).Corollary 0.3. Assume that ϕ satisfies (i) in Theorem 0.1.

Then for all choices of signsξ ∈{−1, 1}G×G (indexed by G × G this time) the product(s, t) →ξ(s, t)ϕ(st)is in ˜V (G, G) hence it defines a Schur multiplier of B(ℓ2(G), ℓ2(G)).Actually, the group structure plays a rather limited role in the preceding statement andin Theorem 0.1. To emphasize this point we state (see also Remark 2.4 below)4

Corollary 0.4. Let G be any set.

Suppose given a mapp : G × G →Gsuch that for all fixed (s0, t0) in G × G the maps s →p(s, t0) and t →p(s0, t) are bijective. (Actually it suffices to assume that there is a fixed finite upper bound on the cardinality ofthe sets {s|p(s, t0) = x} and {t|p(s0, t) = x} when x, s0, t0 run over G).

Let ϕ : G × G →|Cbe a function on G × G. Assume that for all (actually “almost all” is enough) choices ofsigns (εx)x∈G the function(s, t) →εp(s,t)ϕ(p(s, t))is a Schur multiplier of B(ℓ2(G), ℓ2(G)). Then, for all choices of signs εs,t (indexed by G×Gthis time) the function(s, t) →εs,tϕ(p(s, t))is a Schur multiplier of B(ℓ2(G), ℓ2(G)).The results stated above are proved in section 2.

In section 3, we apply them to studya class of “lacunary subsets” of a discrete group which is analogous of the class of finiteunions of Hadamard-lacunary subsets of IN. We give a combinatorial characterization ofthese sets which we call L-sets, but we leave as a conjecture a stronger result (see conjecture3.5 below).5

§ 1. Preliminary Background.We refer to [Pa] for more information on completely bounded maps.Let S be any set.

As usual we denote by ℓ∞(S) the space of all complex valued boundedfunctions on S, equipped with the sup-norm.For any Banach space E, we will also use the space ℓ∞(S, E) of all E-valued boundedfunctions x : S →E equipped with the norm ∥x∥= sups∈S ∥x(s)∥E.When S = IN, we write simply ℓ∞. In particular, we will use below the space ℓ∞(ℓ2)which also can be regarded as the space of all matrices x(j, k) such thatsupj(Xk|x(j, k)|2)1/2 < ∞.Let X, Y be Banach spaces and let X ⊗Y be their linear tensor product.

We recall thedefinition of the projective norm and of several other important tensor norms (cf. [G]).For any u in X ⊗Y , let(1.1)∥u∥∧= inf{nX1∥xi∥∥yi∥|u =nX1xi ⊗yi, xi ∈X, yi ∈Y }We will also need(1.2)γ2(u) = inf{ supξ∈BX∗(nX1|ξ(xi)|2)1/2 supη∈BY ∗(nX1|η(yi)|2)1/2}where the infimum runs again over all possible representations of the form u = Pn1 xi ⊗yi.Equivalently γ2(u) is the “norm of factorization through a Hilbert space” of theassociated operator u : X →Y ∗.We will also need a generalization of the γ2-norm considered in [K] for Banach lattices.Recall that a Banach lattice X is called 2-convex if we have∀x, y ∈X(|x|2 + |y|2)1/2 ≤(∥x∥2 + ∥y∥2)1/2,see e.g.

[LT] for more information.Let X, Y be two 2-convex Banach lattices. For u = Pn1 xi ⊗yi ∈X ⊗Y , we define(1.3)γ(u) = inf{(nX1|xi|2)1/2X(nX1|yi|2)1/2Y}where the infimum runs over all representations of u.6

It is easy to see that the 2-convexity of X and Y implies that this is a norm on X ⊗Y .Note that ℓ∞(or more generally Lp for 2 ≤p ≤∞) is an example of a 2-convex Banachlattice. In the case of the product ℓ∞⊗ℓ∞it is easy to check that (1.2) and (1.3) areidentical, so that(1.4)γ = γ2 on ℓ∞⊗ℓ∞Indeed, if x1, ..., xn ∈ℓ∞(S) over an index set S we have clearlysupξ∈Bℓ∗∞(X|ξ(xi)|2)1/2 =supP|αi|2≤1Xαixiℓ∞= sups∈S(X|xi(s)|2)1/2 = {(X|xi|2)1/2ℓ∞Let S and T be two index sets.

Consider u in ℓ∞(S) ⊗ℓ∞(T) with associated matrixu(s, t) =< δs ⊗δt, u > (we denote by (δs) and (δt) the Dirac masses at s and t respectively,viewed as linear functionals on ℓ∞(S) and ℓ∞(T)). Then we have γ2(u) ≤1 iffthere aremaps x : S →ℓ2y : T →ℓ2such that sups∈S ∥x(s)∥≤1, supt∈T ∥y(t)∥≤1 and∀s, t ∈Tu(s, t) =< x(s), y(t) > .This is very easy to check.The following result is well knownProposition 1.1.

Let S, T be arbitrary sets. Let ϕ : S ×T →|C be a function.

We considerthe Schur multiplierMϕ : B(ℓ2(S), ℓ2(T)) →B(ℓ2(S), ℓ2(T))defined in matrix notation by Mϕ((a(s, t))) = (ϕ(s, t)a(s, t)). The following are equivalent(i) ∥Mϕ∥≤1,(ii) There are vectors x(s), y(t) in a Hilbert space such thatsups ∥x(s)∥≤1, supt ∥y(t)∥≤1 and ϕ(s, t) =< x(s), y(t) >.

(iii) For all finite subsets E ⊂S and F ⊂T we haveX(s,t)∈E×Fϕ(s, t)es ⊗etℓ∞(S) ˆ⊗γ2ℓ∞(T )≤1.Moreover if S and T are finite sets then (i) (ii) and (iii) are equivalent to7

(iv)P(s,t)∈S×T ϕ(s, t)es ⊗etℓ∞(S) ˆ⊗γ2ℓ∞(T ) ≤1 where (es) and (et) denote thecanonical bases of ℓ∞(S) and ℓ∞(T) respectively.Proof. Let us first assume that S and T are finite sets.

The equivalence of (ii) (iii) and (iv)is then obvious. Assume (i).

This means exactly that for any a : ℓ2(S) →ℓ2(T) with ∥a∥≤1and for any α and β in the unit ball respectively of ℓ2(S) and ℓ2(T) we have|Xs,tϕ(s, t)a(s, t)α(s)β(t)| ≤1.In other words ∥Mϕ∥≤1 means that ϕ lies in the polar of the set C1 of all matrices ofthe form (α(s)a(s, t)β(t)) with a, α, β as above. But it turns out that this set C1 is itself thepolar of the set C2 of all matrices (ψ(s, t)) such that ∥P ψ(s, t)es ⊗et∥ℓ∞(S)⊗γ2ℓ∞(T ) ≤1.

(Indeed, this follows from the known factorization property which describes the norm γ∗2which is dual to the norm γ2,cf. e.g.

[Kw] or [P1] chapter 2.b). In conclusion ϕ belongs toC002= C2 iff(i) holds, and this proves the equivalence of (i) and (iv) in the case S and Tare finite sets.In the general case of arbitrary sets S and T, we note that (ii) ⇒(iii) ⇒(i) is obviousby passing to finite subsets.

It remains to prove (i) ⇒(ii), but this is immediate by acompactness argument. Indeed, if (i) holds there is obviously a net (ϕi) tending to ϕpointwise and formed with finitely supported functions on S × T such thatMϕi ≤1.Then by the first part of the proof, each ϕi satisfies (ii) and it is easy to conclude by anultraproduct argument that ϕ also does.Remark.

As observed by Uffe Haagerup (see [H3]) Proposition 1.1 implies that thecompletely bounded norm of Mϕ coincides with its norm. Indeed, it is easy to deduce from(ii) that ∥Mϕ∥cb ≤1.In the harmonic analysis literature, the c.b.

multipliers of Cλ(G) are sometimes calledHerz-Schur multipliers. They were considered by Herz (in a dual framework, as multiplierson A(G)) before the notion of complete boundedness surfaced.

The next result from [BF](see also [H3]) clarifies the relation between the various kinds of multipliers.Proposition 1.2. Let G be a discrete group.

Consider a function ϕ : G →|C. We definethen complex functions ϕ1, ϕ2 and ϕ3 on G × G by setting∀(s, t) ∈G × Gϕ1(s, t) = ϕ(st−1), ϕ2(s, t) = ϕ(s−1t), ϕ3(s, t) = ϕ(st).8

We consider the corresponding Schur multipliers Mϕ1, Mϕ2 and Mϕ3 on B(ℓ2(G), ℓ2(G)).Then(i) ([BF]) The Schur multiplier Mϕ1 is bounded iffthe linear operator Tϕ : C∗λ(G) →C∗λ(G) which maps λ(x) to ϕ(x)λ(x) is completely bounded. Moreover, ∥Mϕ1∥= ∥Tϕ∥cb.

(ii) Moreover, Mϕ1 is bounded iffMϕ2 (resp. Mϕ3) is bounded and we have∥Mϕ1∥= ∥Mϕ2∥= ∥Mϕ2∥cb = ∥Mϕ3∥= ∥Mϕ3∥cb .Proof.

The last assertion is immediate (note that if ψ(s, t) is a bounded Schur multiplieron S × T then for any bijections f : S →S and g : T →Tψ(f(s), g(t)) also is abounded Schur multiplier with the same norm). Note that Mϕ1 leaves C∗λ(G) invariant andits restriction to C∗λ(G) coincides with Tϕ.

Hence by the preceding remark we have∥Tϕ∥cb ≤∥Mϕ1∥cb = ∥Mϕ1∥.Conversely, if ∥Tϕ∥cb ≤1 then the factorization theorem of c.b. maps due to Wittstock(Haagerup [H3] and Paulsen proved it independently, see [Pa]) says that there is a Hilbertspace H, a representation π : B(ℓ2(G)) →B(H) and operators V1 and V2 from ℓ2(G) intoH with ∥V1∥≤1, ∥V2∥≤1 such that ∀a ∈C∗λ(G)Tϕ(a) = V ∗2 π(a)V1.In particular we have ϕ(x)λ(x) = Tϕ(λ(x)) = V ∗2 π(λ(x))V1, which implies∀s, t ∈Gϕ(st−1) =< δs, Tϕ(λ(st−1))δt >=< π(λ(s))∗V2δs, π(λ(t−1))V1δt >This shows that ϕ1 satisfies (ii) in Proposition 1.1, hence ∥Mϕ1∥≤1.Grothendieck [G] proved that γ2 and ∥∥∧are equivalent norms on ℓ∞⊗ℓ∞(or onℓ∞(S) ⊗ℓ∞(T), more precisely there is a constant KG such that(1.5)∀u ∈ℓ∞⊗ℓ∞∥u∥∧≤KGγ2(u).The exact numerical value of the best constant KG in (1.5) is still an open problem (see[P1] for more recent results).Grothendieck’s striking theorem admits many equivalent reformulations.

In the contextof Banach lattices, Krivine [K] emphasized the following one. Let X, Y be 2-convex Banachlattices, then γ and ∥∥∧are equivalent norms on X ⊗Y and we have(1.6)∀u ∈X ⊗Y∥u∥∧≤KGγ(u).9

Note that in (1.5) and (1.6) the converse inequality is trivial (since ∥∥∧is the “greatestcross-norm”), we have γ2(u) ≤∥u∥∧and γ(u) ≤∥u∥∧for all u in X ⊗Y .The reader should note that the equality γ = γ2 on ℓ∞⊗ℓ∞is a special property of ℓ∞spaces. If X = Y = ℓ2 for instance then on X ⊗Yγ2 is the injective norm (i.e.

the usualoperator norm) while γ is identical to the projective norm (i.e. the trace class norm).We refer the reader to [P2] for the discussion of a more general class of cross-normswhich behave like γ and γ2.While the proof of Proposition 1.1 uses nothing more than the Hahn-Banach theorem,the next result is a reformulation of Grothendieck’s theorem one more time, it was observedin some form already in [G] (Prop.

7, p. 68), and was later rediscovered and extended byvarious authors, notably J.Gilbert in harmonic analysis (see [GL],[Be]) and U.Haagerup inoperator algebras (see [H3] the unpublished preliminary version of [H2]).Theorem 1.3. In the case when S, T are finite sets in the same situation as Proposition1.1 we have1KGXϕ(s, t)es ⊗etℓ∞(S) ˆ⊗ℓ∞(T ) ≤∥Mϕ∥≤Xϕ(s, t)es ⊗etℓ∞(S) ˆ⊗ℓ∞(T ) .Moreover, when S, T are arbitrary sets, the space of all bounded Schur multipliers ofB(ℓ2(S), ℓ2(T)) coincides with the space ˜V (S, T).Proof.

This follows immediately from Grothendieck’s inequality (1.5) and Proposition 1.1.Perhaps a more intuitive formulation is as follows. Let us call “simple multipliers” theSchur multipliers of the formϕ(s, t) = εsηtwith εs, ηt ∈|C such that |εs| ≤1, |ηt| ≤1.

These are obviously such that ∥Mϕ∥≤1, butprecisely Theorem 1.3 says that any multiplier ϕ with ∥Mϕ∥≤1KG lies in the convex hullof the set of simple multipliers if S, T are finite sets and if S, T are infinite sets, then ϕ liesin the pointwise closure of the convex hull of the set of simple multipliers.Remark 1.4 :Consider again a “simple Schur multiplier” of the form ϕ(s, t) = εsηt asabove with |εs| ≤1, |ηt| ≤1. Then we have∀A ∈B(ℓ2(S), ℓ2(T))Mϕ(A) = v1Av210

where v1 : ℓ2(S) →ℓ2(S) and v2 : ℓ2(T) →ℓ2(T) are the diagonal operators of multiplicationby (εs) and (ηt) respectively. Therefore, it is obvious that ∥Mϕ∥cb ≤1.

(see [Pa] for moreinformation. )We need to consider “sums” of Banach spaces which are usually not direct sums.Although we will mainly work with natural concrete Banach spaces X and Y for whichsaying that an element belongs to X + Y will have a clear meaning, we recall the followingformal definition of X + Y .Assume that X, Y are both continuously injected in a larger topological vector spaceX .

Then X + Y is defined as the subspace of X of all elements of the form σ = x + y withx ∈X, y ∈Y , equipped with the norm∥σ∥X+Y = inf{∥x∥X + ∥y∥Y |σ = x + y}.Equipped with this norm, X + Y is a Banach space (and its dual can be identified withX∗∩Y ∗under some mild compatibility assumption on X, Y ).Alternately, one may consider the direct sum X ⊕1Y equipped with the norm ∥(x, y)∥=∥x∥+∥y∥together with the closed subspace N ⊂X ⊕1 Y of all elements (x, y) which satisfythe identity x + y = 0 when injected into X .Then the quotient space P = (X ⊕1 Y )/N can be identified with X + Y .It will be convenient at some point to use the following elementary fact.Lemma 1.5. Let dm(t) = dt2π be the normalized Haar measure on T. Then for any integerN and any continuous function f : TN →IR we haveZf(t1, ..., tN)dm(t1)...dm(tN) ≥infZf(ein1t, ein2t, ..., einNt)dm(t)where the infimum on the right side runs over all sets of integers n1, n2, ..., nN with2n1 < n2, 2n2 < n3, ..., 2nN−1 < nN.Proof.

If f is a trigonometric polynomial, this is obvious by choosing (nk) lacunary enough.By density, this must remain true for all real valued f in C(TN). Let us consider the infinitedimensional torus TIN.

We denote by zj the j-th coordinate on TIN and by µ the normalizedHaar measure on TIN. The following is a reformulation of the main result of [LPP].11

Theorem 1.6. Let a1, ..., an be elements of a von Neumann algebra M, let ξ1, ..., ξn beelements of the predual M∗.

Then(1.7)|nX1< ξj, aj > | ≤Z XzjajM∗dµ(z)[(Xa∗jaj)1/2M +(Xaja∗j)1/2M].Proof. Two approaches are given in [LPP].

The first one proves this result using thefactorization of analytic functions in H1 with values in M∗. Actually, in [LPP] (1.7) is statedwithR∥P zjξj∥M∗dµ(z) replaced byR P einjtξjM∗dm(t) for any lacunary sequence njsuch that nj > 2nj−1.

Using the preceding lemma, it is then easy to obtain (1.7) as statedabove. (Moreover, it is possible to use the factorization argument of [LPP] directly in TN,see the following remark.) A second approach is given in the appendix of [LPP].

There itis shown that (1.7), with some additional numerical factor, can be deduced from (and isessentially equivalent to) the non-commutative Grothendieck inequality due to the author(see [P1], Theorem 9.4 and Corollary 9.5).Remark.The reader may find the use of a lacunary sequence (nk) in the preceding proof a bitartificial. Actually, we can use directly the independent sequence (zk) on TIN equipped withµ.

Indeed, the classical factorization theory of H1 functions as products of two H2 functionsextends to this setting, provided one considers TIN as a compact group with ordered dualin the sense e.g. of [R] chapter 8.

Here the dual of TIN is ordered lexicographically. Thefactorization of matrix valued functions (as used in [P2] Appendix B) also extends to thissetting, so that the main results of [P2] also remain valid in this setting.

This approach isdescribed in [P3]. We chose the more traditional “one dimensional” torus presentation toprovide more precise and explicit references for the reader.We will use the following well known consequence of the Hahn-Banach Theorem (cf.

[Kw], see also e.g. Lemma 1.3 in [P2]).Lemma 1.7.

Let S, T be finite sets. Let u : ℓ∞(S, ℓ2)ˆ⊗γℓ∞(T) →|C be a linear form ofnorm ≤1 on ℓ∞(S, ℓ2)ˆ⊗γℓ∞(T).

Then there are probabilities P, Q on S and T such that∀ϕ ∈ℓ∞(S, ℓ2)∀η ∈ℓ∞(T). (1.8)| < u, ϕ ⊗η > | ≤(Z∥ϕ(s)∥2ℓ2 dP(s))1/2(Z|η(t)|2dQ(t))1/2.Proof.

(Sketch). By assumption and by definition (1.3) we have for all finite sequences12

ϕk ∈ℓ∞(S, ℓ2), ηk ∈ℓ∞(T)Xk| < u, ϕk ⊗ηk > | ≤sups∈S(X∥ϕk(s)∥2)1/2 supt∈T(X|ηk(t)|2)1/2≤12 supS×T{X∥ϕk(s)∥2 + |ηk(t)|2}Let C be the convex cone in C(S × T) formed by all the functions of the form(s, t) →12X∥ϕk(s)∥2 + |ηk(t)|2 −| < u, ϕk ⊗ηk > |.Then C is disjoint from the open cone C−= {ϕ| max ϕ < 0}, hence (Hahn-Banach) there isa hyperplane in C(S ×T) which separates C and C−. By an obvious adjustment, this yieldsa probability λ on S × T such thatRf(s, t)dλ ≥0 for any f in C. Hence letting P (resp.Q) be the projection of λ on the first (resp.

second) coordinate we obtain| < u, ϕ ⊗η > | ≤12(Z∥ϕ(s)∥2 dP(s) +Z|η(t)|2dQ(t)).Finally applying this to (ϕθ−1) ⊗(θη) and minimizing the right hand side over all θ > 0, weobtain the announced result (1.8).Remark 1.8 Let S, T be finite sets and let P, Q be probabilities on S and T respectively.Let JP : ℓ∞(S) →L2(P) and JQ : ℓ∞(T) →L2(Q) be the canonical inclusions. Then wehave ∀ψ ∈ℓ∞(S) ⊗ℓ∞(T)(1.9)∥(JP ⊗JQ)(ψ)∥L2(P ) ˆ⊗L2(Q) ≤∥ψ∥ℓ∞(S) ˆ⊗γ2ℓ∞(T ) .Indeed, this is elementary.

For any x1, ..., xn in ℓ∞(S), y1, ..., yn in ℓ∞(T), we haveX∥xi∥2L2(P ))1/2 ≤∥(X|xi|2)1/2∥ℓ∞(S)andX∥yi∥2L2(P ))1/2 ≤∥(X|yi|2)1/2∥ℓ∞(T ).This clearly implies (1.9).Remark 1.9 Let us denote simply by H1(T; M∗) the subspace of L1(T, dm; M∗) formed ofall the functions f such that the (M∗-valued ) Fourier transform is supported on the non-negative integers. Similarly, we can denote by H1(TIN; M∗) the subspace of L1(TIN, mIN; M∗)formed by the functions with Fourier transform supported by the non-negative elements ofZZ(IN) ordered lexicographically.

We again denote by zj the j-th coordinate on TIN and welet ˆf(zj) =Rf ¯zj. In [LPP] the following refinement of (1.7) is proved.Assume that there is a function f in the unit ball of H1(T; M∗) (resp.

H1(TIN; M∗))such that ˆf(3j) = aj (resp. ˆf(zj) = aj) for all j , then we have(1.10)|nX1< ξj, aj > | ≤[(Xa∗jaj)1/2M +(Xaja∗j)1/2M].13

§ 2. Main results.Our main result is a general statement which does not use the group structure at all,it can be viewed as a generalization of Varopoulos’s result stated above as Theorem 0.2.Theorem 2.1.

Let S and T be arbitrary sets, and letψn ∈ℓ∞(S)ˆ⊗ℓ∞(T)be a sequence such that the series∞Xn=1εnψnconverges in ℓ∞(S)ˆ⊗ℓ∞(T) for almost all choice of signs εn = ±1. Then, if we denote by(en) the canonical basis of ℓ2, the series∞Xn=1en ⊗ψnis convergent in the space ℓ∞(S, ℓ2)ˆ⊗ℓ∞(T) + ℓ∞(S)ˆ⊗ℓ∞(T, ℓ2).Note.

The spaces ℓ∞(S, ℓ2)ˆ⊗ℓ∞(T) and ℓ∞(S)ˆ⊗ℓ∞(T, ℓ2) are naturally continuouslyinjected into ℓ∞(S × T, ℓ2), which is used to define the above sum.Notation. Let Ω= TIN.

Let µ be the normalized Haar measure on Ω, i.e. µ = ( dt2π)IN.We denote by z = (zk)k∈IN a generic point of Ω(and we consider the k −th coordinate zkas a function of z).We will denote ℓ∞instead of ℓ∞(IN) and ℓ∞(ℓ2) instead of ℓ∞(IN, ℓ2).With this notation, we can state a more precise version of Theorem 2.1.Theorem 2.2.

In the same situation as Theorem 2.1, let ψ1, ..., ψn be a finite sequence inℓ∞(S)ˆ⊗ℓ∞(T). (i) Assume(2.1)Zγ2(nX1zkψk)dµ(z) < 1.Then there is a decomposition in ℓ∞(S)ˆ⊗ℓ∞(T) of the formψk = Ak + Bk14

such that (with γ as defined in (1.3))nX1ek ⊗Akℓ∞(S,ℓ2) ˆ⊗γℓ∞(T )< 1andnX1ek ⊗Bkℓ∞(S) ˆ⊗γℓ∞(T,ℓ2)< 1(ii) AssumeZ nX1zkψkℓ∞(S) ˆ⊗ℓ∞(T )dµ(z) < 1then there is a decomposition ψk = Ak + Bk such thatXek ⊗Akℓ∞(S,ℓ2) ˆ⊗ℓ∞(T ) < KG andXek ⊗Bkℓ∞(S) ˆ⊗ℓ∞(T,ℓ2) < KG,where KG is the Grothendieck constant.Proof. The proof is based on the main result of [LPP] reformulated above as Theorem 1.6.By a standard Banach space technique, Theorem 2.2 can be reduced to the case when S andT are finite sets.

(Use the fact that ℓ∞is a L∞space, more precisely it can be viewed as theclosure of the union of an increasing family of finite dimensional sublattices each isometricto ℓ∞(S) for some finite set S).We will denote by α1 (resp. α2) the norm on ℓ1(S, ℓn2) ⊗ℓ1(T) (resp.

ℓ1(S) ⊗ℓ1(T, ℓn2))which is dual to the norm in ℓ∞(S, ℓn2)ˆ⊗γℓ∞(T) (resp. ℓ∞(S)ˆ⊗γℓ∞(T, ℓn2)).Let (ek) be the canonical basis of ℓn2.

Let Ak ∈ℓ1(S) ⊗ℓ1(T) and let Φ = P ek ⊗Ak.We will make the obvious identifications permitting to view Φ as an element either ofℓ1(S, ℓn2) ⊗ℓ1(T) or of ℓ1(S) ⊗ℓ1(T, ℓn2). Then, by duality Theorem 2.2 (i) is equivalentto the following inequality.For all ψk in ℓ∞(S)ˆ⊗γ2ℓ∞(T)(2.2)|X< Ak, ψk > | ≤Zγ2(Xzkψk)dµ(z)[α1(Φ) + α2(Φ)].To check this, by homogeneity we may assume (2.1) and also(2.3)α1(Φ) + α2(Φ) = 1.15

Then, by Lemma 1.7, there are probabilities P1, P2 on S and Q1, Q2 on T such that(with obvious identifications).∀ϕ ∈ℓ∞(S, ℓn2)∀η ∈ℓ∞(T)| < ϕ ⊗η, Φ > | ≤α1(Φ)Z∥ϕ(s)∥2ℓn2 dP1(s)Z|η(t)|2dQ1(t)1/2and∀β ∈ℓ∞(S)∀ω ∈ℓ∞(T, ℓn2)| < β ⊗ω, Φ > | ≤α2(Φ)Z|β(s)|2dP2(s)Z∥ω(t)∥2ℓn2 dQ2(t)1/2.Now let P= α1(Φ)P1 + α2(Φ)P2, Q = α1(Φ)Q1 + α2(Φ)Q2. By (2.3) these areprobabilities.Then(2.4)| < ϕ ⊗η, Φ > | ≤Z∥ϕ(s)∥2ℓn2 dP(s)1/2 Z|η(t)|2dQ(t)1/2and(2.5)| < β ⊗ω, Φ > | ≤Z|β(s)|2dP(s)1/2 Z∥ω(t)∥2ℓn2 dQ(t)1/2.This means that Ak defines a bounded linear operator ak : L2(P) →L2(Q)∗such that< ak(β), η >=< β ⊗η, Ak >.

Moreover (2.5) and (2.4) imply respectively ∥P a∗kak∥≤1and ∥P aka∗k∥≤1. Let JP : ℓ∞(S) →L2(P) and JQ : ℓ∞(T) →L2(Q) be the canonicalinclusions.

Let ξk = (JP ⊗JQ)(ψk) ∈L2(P) ⊗L2(Q). By (1.9) we haveZ XzkξkL2(P ) ˆ⊗L2(Q) dµ(z) < 1.Note that < ξk, ak >=< ψk, Ak >.

Hence applying (1.7) we obtain the desired inequality(2.2). This concludes the proof of the first part.

The second part is an immediate consequenceof the first one by (1.6).Remark : It is also possible to deduce Theorem 2.2 directly from the factorization Theoremof [P2] (see Corollary 1.7 or Theorem 2.3 in [P2]), which applies in particular to functionsin H1 with values in ℓ∞(S)ˆ⊗γ2ℓ∞(T). Using this, the argument of [LPP] then gives the16

decomposition of Theorem 2.2 in a somewhat more explicit fashion as a formula in terms ofthe factorization of the “analytic” function z →P zkψk.Remark 2.3. Let Ak be as above such that(2.9)nX1ek ⊗Akℓ∞(S,ℓ2) ˆ⊗γℓ∞(T )< 1.Then for any n-tuple t1, ..., tn in T we have(2.10)sups∈S nXk=1|Ak(s, tk)|2!1/2< 1.A similar remark holds for Pn1 ek ⊗Bk.Indeed, by the definition (1.3), (2.9) means that there is a Hilbert space H and elementsα in ℓ∞(S, ℓ2(H)) and β in ℓ∞(T, H) each with norm < 1 such that(2.11)∀k = 1, ..., nAk(s, t) =< αk(s), β(t) >(where α(s) ∈ℓ2(H) and αk(s) denotes the k −th coordinate of α(s)).

Then (2.10) is animmediate consequence of (2.11).We now derive Theorem 0.1 from Theorems 2.1 and 2.2.Proof of Theorem 0.1. (i) ⇒(ii) is trivial.

Assume (ii). By Remark 1.2 for almost all choices of signsε in {−1, 1}G the function (s, t) →ε(st)ϕ(st) defines a c.b.

Schur multiplier Mεϕ ofB(ℓ2(G), ℓ2(G)).We can assume ∥Mεϕ∥cb ≤F(ε) for some measurable function F(ε) finite almosteverywhere on {−1, 1}G. A fortiori for each finite subsets S ⊂G and T ⊂G, the function(s, t) →ε(st)ϕ(st) restricted to S ×T is a c.b. Schur multiplier of B(ℓ2(S), ℓ2(T)) with norm≤F(ε).

By Proposition 1.1, this means that we have for all finite subsets S ⊂G, T ⊂GX(s,t)∈S×Tε(st)ϕ(st)es ⊗etℓ∞(S) ˆ⊗γ2ℓ∞(T )≤F(ε),where we have denoted by (es) and (et) the canonical bases of ℓ∞(S) and ℓ∞(T).Equivalently we have(2.12)Xx∈STε(x)ϕ(x)X(s,t)∈S×Tst=xes ⊗etℓ∞(S) ˆ⊗γ2ℓ∞(T )≤F(ε).17

By a classical integrability result of Kahane (cf. [Ka]) we can assume that F is integrableover {−1, 1}G, so that there is a number C > 0 such that the average over ǫ of the left side of(2.12) is less than C. By a simple elementary reasoning (decompose into real and imaginaryparts, use the triangle inequality and the unconditionality of the average over ǫ), it followsfrom (2.12) that if µG denotes the normalized Haar measure on TG and if z = (zx)x∈Gdenotes a generic point of TG, we haveZXx∈STzxϕ(x)X(s,t)∈S×Tst=xes ⊗etℓ∞(S) ˆ⊗γ2ℓ∞(T )dµ(z) < 2C.Let ψx = ϕ(x) P(s,t)∈S×Tst=xes ⊗et for all x in ST and let ψx = 0 otherwise.Then by Theorem 2.2 and Remark 2.3 we have a decompositionψx = Ax + Bx in ℓ∞(S) ⊗ℓ∞(T)such thatsups∈S Xt∈T|Ast(s, t)|2!1/2< 2Candsupt∈T Xs∈S|Bst(s, t)|2!1/2< 2CThis yields functions ϕ1 and ϕ2 on S × T such that ϕ1(s, t) = Ast(s, t), ϕ2(s, t) = Bst(s, t)and∀(s, t) ∈S × Tϕ(st) = ψst(s, t) = ϕ1(s, t) + ϕ2(s, t).Hencesups∈S Xt∈T|ϕ1(s, t)|2!1/2< 2Csupt∈TX|ϕ2(s, t)|21/2< 2C.Let us denote by ϕS,T1and ϕS,T2the functions obtained on G × G by extending ϕ1 andϕ2 by zero outside S × T.Now if we let S × T tend to G × G along the set of all products of finite setsdirected by inclusion and if we let Φ1, Φ2 be pointwise cluster points of the correspondingsets (ϕST1 ) and (ϕS,T2), we obtain finally two functions Φ1 and Φ2 on G × G such thatϕ(st) = Φ1(s, t) + Φ2(s, t) for all (s, t) in G × G and satisfyingsups∈G Xt∈G|Φ1(s, t)|2!1/2≤2Csupt∈G Xs∈G|Φ2(s, t)|2!1/2≤2C18

Let then Γ1 ∪Γ2 = G × G be a partition defined byΓ1 = {(s, t) ∈G × G||Φ1(s, t)| ≥|Φ2(s, t)|}Γ2 = {(s, t) ∈G × G||Φ1(s, t)| < |Φ2(s, t)|}It is then clear that ϕ satisfies the property (iii) in Theorem 0.1. This shows (ii) ⇒(iii).The equivalence (iii) ⇔(iv) is part of Theorem 0.2 (due to Varopoulos).

(Note that theimplication (iii) ⇒(i) also follows from the implication (iii) ⇒(i) in Varopoulos’s Theorem0.2. )We now show (iii) ⇒(v).

Assume (iii). Let a(x) be as in (v) and let g and h be in theunit ball of ℓ2(G, H).

Assume(2.14)maxXa(x)∗a(x)1/2 ,Xa(x)a(x)∗1/2≤1.It clearly suffices to show that(2.15)|Xs,t∈G×Gϕ(st−1) < h(s), a(st−1)g(t) > | ≤2C.Let Σ1 (resp. Σ2) be the left side of (2.14) with the summation restricted to (s, t−1) ∈Γ1(resp.

(s, t−1) ∈Γ2). Observe that P(s,t−1)∈Γ1 |ϕ(st−1)|2∥h(s)∥2 ≤C2, hence by Cauchy-Schwarz and (2.14)|Σ1| ≤C(X(s,t−1)∈Γ1∥a(st−1)g(t)∥2)1/2 ≤C(X(s,t)∈G×G< a(st−1)∗a(st−1)g(t), g(t) >)1/2≤C(Xt∥g(t)∥2∥Xsa(st−1)∗a(st−1)∥)1/2 ≤C.A similar argument yields |Σ2| ≤C hence (2.15) follows and the proof of (iii) ⇒(v) iscomplete.Finally we show (v) ⇒(i).

We start by recalling that for any finitely supported functiona : G →B(H) we have the elementary inequality(2.16)maxXa(x)∗a(x)1/2 ,Xa(x)a(x)∗1/2≤Xλ(x) ⊗a(x)B(ℓ2(G,H)) .Now assume (v). We have then by (2.16) if supx |ε(x)| ≤1Xx∈Gε(x)ϕ(x)λ(x) ⊗a(x) ≤CXx∈Gλ(x) ⊗a(x) ,19

hence the multiplier of C∗λ(G) defined by εϕ is completely bounded with norm ≤C. Thisproves (v) ⇒(i).Proof of Corollary 0.4.

Let p and ϕ be as in Corollary 0.4.Then there is a constant C such that for all finite subsets S, T of G we haveZ Xx∈p(S,T )ϕ(x)zx(Xp(s,t)=xes ⊗et)ℓ∞(G) ˆ⊗ℓ∞(G)< C.Reasoning as above in the proof of (ii) ⇒(iii) in Theorem 0.1, we find a decomposition ofthe formϕ(p(s, t)) = Ap(s,t)(s, t) + Bp(s,t)(s, t)and using Remark 2.3 and the bounds(2.17)sups,x |{t| p(s, t) = x}| < ∞and supt,x |{s| p(s, t) = x}| < ∞we can obtain that for some constant C′sups∈SXt∈T|Ap(s,t)(s, t)|2 ≤C′supt∈TXs∈S|Ap(s,t)(s, t)|2 ≤C′.We then conclude the proof as in the proof of Theorem 0.1 by a pointwise compactnessargument, showing that (s, t) →ϕ(p(s, t)) satisfies (iii)’ in Theorem 0.2. hence (recallProposition 1.1 or Theorem 1.3) for all bounded complex functions (s, t) →εs,t the function(s, t) →εs,tϕ(p(s, t)) is a Schur multiplier of B(ℓ2(G), ℓ2(G)).Remark 2.4. Let S, T, X be arbitrary sets and let p : S × T →X be a map satisfying(2.17).

Consider a function ϕ : X →|C and let ψ : S × T →|C be defined byψ(s, t) = ϕ(p(s, t)).Let K(T) be the linear span of the canonical basis of ℓ2(T). For any x in X, let Λ(x) :K(T) →|CS be the operator defined by the matrix Λx(s, t) defined by Λx(s, t) = 1 ifp(s, t) = x and Λx(s, t) = 0 otherwise.

Then we can generalize Varopoulos’s theorem asfollows. The following are equivalent20

(i) For all bounded functions ε : S × T →|C the pointwise product εψ is a boundedSchur multiplier of B(ℓ2(S), ℓ2(T). (ii) For almost all choices of signs ε ∈{−1, 1}S×T, the product εψ is a bounded Schurmultiplier of B(ℓ2(S), ℓ2(T).

(iii) There is a partition of S × T say S × T = Γ1 ∪Γ2 such thatsups∈SXt∈T|ψ(s, t)|21{(s,t)∈Γ1} < ∞supt∈TXs∈S|ψ(s, t)|21{(s,t)∈Γ2} < ∞. (iv) There is a constant C such that for any Hilbert space H and for any finitelysupported function a : X →B(H) we haveXx∈Xϕ(x)Λ(x) ⊗a(x)B(ℓ2(S,H),ℓ2(T,H))≤C maxXa(x)∗a(x)1/2 ,Xa(x)a(x)∗1/2.This statement is proved exactly as above.

This applies in particular when p is the productmap on a semigroup. When S = T = X = IN and p(s, t) = s + t we recover results alreadyobtained in [B2].21

§ 3. Lacunary sets.The study of “thin sets” such as Sidon sets in discrete non Abelian groups has beendevelopped by several authors, namely Leinert [L1,L2] Bo ˙zejko ([B1, 2, 3]), Figa-Talamancaand Picardello [FTP] and others.

(See [LR] for the theory of Sidon sets in Abelian groups).In this section, we apply the preceding results to a class of “lacunary” sets which wecall “L-sets”. There is a striking analogy between the “L-sets” defined below and a class ofsubsets of IN which we will call Paley sets.

A subset Λ ⊂IN will be called a Paley set if thereis a constant C such that for all f = P∞n=0 aneint in H1 we have(3.1) Xn∈Λ| ˆf(n)|2!1/2≤C ∥f∥1 .It is well known (cf. e.g.

[R]) that Paley sets are simply the finite unions of Hadamard-lacunary sequences, i.e. of sequences {nk} such that lim infk→∞(nk+1/(nk) > 1.Equivalently, Λ is a Paley set iffthere is a constant C such that∀n > 0|Λ ∩[n, 2n[| ≤C.In [LPP], it is proved that if Λ ⊂IN is a Paley set there is a constant C such that for anyf in H1(H ˆ⊗H) there is a decomposition in H ˆ⊗H of the form ˆf(n) = a(n) + b(n)∀n ∈Λsuch that(3.2)tr Xn∈Λa(n)∗a(n)!1/2+ tr Xn∈Λb(n)b(n)∗!1/2≤C ∥f∥H1(H ˆ⊗H)Moreover, when ˆf is supported by Λ this inequality becomes an equivalence.The papers [LPP] and [HP] suggest that there is a strong analogy between Paleysequences and free subsets of a discrete group G. To explain this we introduce more notation.Let H be a Hilbert space.

We denote by A(G, H ˆ⊗H) the set of all functions f : G →H ˆ⊗Hsuch that for some g, h in ℓ2(G, H) we have∀x ∈Gf(x) =Xst=xg(s) ⊗h(t).Let∥f∥A(G,H ˆ⊗H) = inf{∥g∥ℓ2(G,H) ∥h∥ℓ2(G,H)}22

where the infimum runs over all possible representations. Then (see [HP]) if G is the freegroup on n generators g1, ..., gn, we have the following analogue of (3.1).

For any f inA(G, H ˆ⊗H) then there is a decomposition f(gk) = ak + bk in H ˆ⊗H such that(3.3)tr(Xa∗kak)1/2 + tr(Xbkb∗k)1/2 ≤2 ∥f∥A(G,H ˆ⊗H) .Moreover, when f is supported by Λ this inequality becomes an equivalence.This motivated the followingDefinition 3.1. A subset Λ of a discrete group G will be called an L-set if there is aconstant C such that for any H and for any f in A(G, H ˆ⊗H) we haveinff(x)=a(x)+b(x)tr Xx∈Λa(x)∗a(x)!1/2+ trXb(x)b(x)∗1/2≤C ∥f∥A(G,H ˆ⊗H) ,where the infimum runs over all possible decompositions f(x) = a(x) + b(x) in H ˆ⊗H.Proposition 3.2.

The following properties of a subset Λ ⊂G are equivalent. (i) Λ is an L-set.

(ii) There is a constant C such that for any Hilbert space H and for any finitelysupported function a : Λ →B(H) we haveXx∈Λλ(x) ⊗a(x)B(ℓ2(G,H))≤C maxXa(x)∗a(x)1/2 ,Xa(x)a(x)∗1/2. (iii) For any bounded sequence ε in ℓ∞(G) supported by Λ, the associated multiplierdefined on C∗λ(G) byXf(x)λ(x) →Xf(x)ε(x)λ(x)is completely bounded on C∗λ(G).Proof.

(i) and (ii) are clearly equivalent. They are but a dual reformulation of each other.

(ii) and (iii) are equivalent by Theorem 0.1 applied to the indicator function of Λ.Remark. By Theorem 0.1, the preceding properties are also equivalent to the property(iii)’ obtained by requiring that the property (iii) holds only for almost all choice of signs(ε(x))x∈Λ in {−1, 1}Λ.23

Remark. The preceding result shows that Λ is an L-set iffΛ is a strong 2-Leinert set inthe sense of Bo ˙zejko [B1].

Leinert [L1,L2] first constructed infinite sets of this kind in freenoncommutative groups. Leinert’s results were clarified in [AO].

Moreover, in [H1] severalrelated important inequalities were obtained for the operator norm of the convolution on thefree group by a function supported by the words of a given fixed lenghth in the generators.In particular, it was known to Haagerup (see [HP]) that any free subset of a discrete groupis an L-set. For instance the generators (or the words of length one) on the free group withcountably many generators form an L-set.

On the other hand it is rather easy to see thatthe set of words of a fixed length k > 1 is not an L-set (for instance it clearly does notsatisfy (ii) in Theorem 3.3).The main result of this section is the followingTheorem 3.3. Let G be an arbitrary discrete group.

Let Λ ⊂G be a subset. Let RΛ ⊂G×Gbe defined byRΛ = {(s, t) ∈G × G| st ∈Λ}.The following properties of Λ are equivalent(i) Λ is an L-set.

(ii) There is a constant C such that for any finite subsets E, F ⊂G with |E| = |F| = Nwe have|RΛ ∩(E × F)| ≤CN. (iii) There is a constant C and there is a partition RΛ = Γ1 ∪Γ2 such thatsups∈GXt∈G1(s,t)∈Γ1 ≤Candsupt∈GXs∈G1(s,t)∈Γ2 ≤C.Proof : This is an immediate consequence of Theorem 0.1 applied to the indicatorfunction of Λ.Remark.

As already mentioned, the equivalence between (ii) and (iii) is due to Varopoulos[V1]. Our result shows that Λ ⊂G is an L-set iffRΛ determines a V -set for ℓ∞(G)ˆ⊗ℓ∞(G)in the sense of Varopoulos (see [LP] and [V2]).

Equivalently, let G′ = TG and let (γs)s∈G bethe coordinates on G. Then Λ is an L-set in G iffthe set {γs × γt|st ∈Λ} is a Sidon set inthe dual of G′ × G′ i.e. in ZZ(G) × ZZ(G).24

Remark. Taking Remark 2.4 into account, we can extend the notion of L-set to the casewhen Λ is a subset of a semi-group G embeddable into a group (for instance G = IN).

Inthat case we will say that Λ is an L-set if it satisfies the equivalent properties of Theorem3.3 (or the analogue of (ii) in Proposition 3.2). This provides a common framework for Paleysets and L-sets.

Note however that all the L-subsets of ZZ (or of any amenable group) arefinite, while the L-subsets of IN are exactly the Paley sets. Thus this notion depends of thechoice of the semi-group containing Λ.

If we remain in the category of groups this difficultydoes not arise, if H is a subgroup of a group G and if Λ ⊂H, then Λ is an L-set in H iffitis an L-set in G (this is easy to check e.g. by Proposition 3.2).Note that L-sets are clearly stable under finite unions.

Moreover the translate of anL-set is again an L-set. The only known examples of L-sets seem to be finite unions oftranslates of free sets.

The sets which are translates of a free set (more precisely translatesof a free set augmented by the unit element) are characterized in [A0] as those which havethe Leinert property. In analogy with Paley sequences we formulate the following.Conjecture 3.5.

Every L-set Λ can be written as a finite union x1F1 ∪...∪xnFn wherex1, ..., xn ∈G and F1, ..., Fn are free subsets of G.(Here, the subset reduced to the unitelement is considered free, so that a singleton is a translate of a free set. )It is possible to check that if Λ satisfies (iii) in Theorem 3.3 with C = 1 then it satisfiesthe Leinert property in the sense of [AO], hence it is a translate of a free set augmented bythe unit element, a fortiori it is the union of two translates of free sets.

Therefore to verifythe above conjecture it suffices to prove that any set Λ satisfying (iii) in Theorem 3.3 withsome constant C can be written as a finite union of sets satisfying the same property withC = 1.25

§ 4. A more general framework.Actually, Theorem 2.2 can be extended to a rather general situation already consideredin [P2].

We describe this briefly since it is easy to adapt the preceding ideas to this setting.Let X be a Banach space. We will identify an element u in X ⊗ℓ2 with an operatoru : X∗→ℓ2 (of finite rank and weak-* continuous).

Hence, for any ξ in X∗, u(ξ) ∈ℓ2.A norm δ on X ⊗ℓ2 will be called 2-convex if there is a constant c > 0 such that forany u in X ⊗ℓ2(4.1)c∥u∥= csupξ∈X∗,∥ξ∥≤1∥u(ξ)∥≤δ(u)and such that for all u, u1, u2 in X ⊗ℓ2 satisfying∀ξ ∈X∗∥u(ξ)∥≤∥u1(ξ)∥2 + ∥u2(ξ)∥21/2,we have(4.2)δ(u) ≤(δ(u1)2 + δu2)21/2 .Note that if ∥u(ξ)∥= ∥u1(ξ)∥for all ξ in X∗, we must have δ(u) = δ(u1), moreover for allT : ℓ2 →ℓ2 we have δ(Tu) ≤∥T∥δ(u).Now let X, Y be two Banach spaces and let δ1 (resp. δ2) be a 2-convex norm on X ⊗ℓ2(resp.

Y ⊗ℓ2). We can introduce a norm Γ on X ⊗Y by setting ∀u ∈X ⊗Y, u = Pni=1 xi⊗yiΓ(u) = inf(δ1(nXi=1xi ⊗ei)δ2(nXi=1yi ⊗ei))where the infimum runs over all possible decompositions of u and where ei denotes thecanonical basis of ℓ2.

It is easy to see that this is a norm. We denote by X ˆ⊗ΓY the completionof X ⊗Y for this norm.We also denote by X ˆ⊗δ1ℓ2 and Y ˆ⊗δ2ℓ2 the completions of X ⊗ℓ2 and Y ⊗ℓ2 for thenorms δ1 and δ2.Assume that X and Y are continuously injected in a Banach space Z.

Then, by(4.1) X ˆ⊗δ1ℓ2 and Y ˆ⊗δ2ℓ2 are both continuously injected into Z∨⊗ℓ2 (the injective tensorproduct), so that we can give a meaning to the sum X ˆ⊗δ1ℓ2 + Y ˆ⊗δ2ℓ2.For simplicity, we denoteX[ℓ2] = X ˆ⊗δ1ℓ2 and Y [ℓ2] = Y ˆ⊗δ2ℓ2.26

We can now equip X[ℓ2] ⊗ℓ2 with a 2-convex norm ∆1 as follows. For any v = Pni=1 vi ⊗eiwith vi ∈X[ℓ2] there is clearly an operator w ∈X ˆ⊗δ1ℓ2 (not necessarily unique) such that∥w(ξ)∥=Pni=1 ∥vi(ξ)∥21/2∀ξ ∈X∗.

We then define∆1(v) = δ1(w).By (4.2) this does not depend on the particular choice of w. Clearly, this defines (by thedensity of ∪ℓn2 in ℓ2) a 2-convex norm ∆1 on X[ℓ2]⊗ℓ2. Now using the pair (∆1, δ2) (insteadof the pair (δ1, δ2)) we can define the spaceX[ℓ2]ˆ⊗ΓYexactly as above for X ˆ⊗ΓY .Similarly, we can define ∆2 on Y [ℓ2] ⊗ℓ2 and using the pair (δ1, ∆2) we construct thespaceX ˆ⊗ΓY [ℓ2]exactly as above for X ˆ⊗ΓY .Assume that X, Y are both continuously injected in a Banach space Z.

Then, by (4.1),it is easy to check that X ˆ⊗δ1ℓ2 and Y ˆ⊗δ2ℓ2 are both continuously injected into the injectivetensor products X∨⊗ℓ2 and Y∨⊗ℓ2, and consequently also X[ℓ2]ˆ⊗ΓY and X ˆ⊗ΓY [ℓ2] areboth continuously injected into X∨⊗Y∨⊗ℓ2, so that using this inclusion we may considerthe sumX[ℓ2]ˆ⊗ΓY + X ˆ⊗ΓY [ℓ2],with its natural norm (see section 1).Then Theorems 2.1 and 2.2 can be generalized as follows.Theorem 4.1. With the preceding notation, consider elements ψn in X ˆ⊗ΓY such that theseries P∞n=1 εnψn converges for almost all choice of signs εn = ±1.

Then necessarily theseries P∞n=1 en ⊗ψn converges in the space X[ℓ2]ˆ⊗ΓY + X ˆ⊗ΓY [ℓ2].Theorem 4.2. Let ψ1, ..., ψN be a finite sequence in X ˆ⊗ΓY such thatZΓ(NX1zkψk)dµ(z) < 1.27

Then there is a decomposition in X ˆ⊗ΓY of the formψk = Ak + Bksuch thatNX1ek ⊗AkX[ℓ2] ˆ⊗ΓY< 1andNX1ek ⊗BkX ˆ⊗ΓY [ℓ2]< 1. ( Note.

Here of course P ek ⊗Ak is identified with an element of X[ℓ2]ˆ⊗ΓY in theobvious natural way and similarly for P ek ⊗Bk. )The proof is the same as for Theorems 2.1 and 2.2.

We leave the details to the reader.Remark 4.3. In particular, with the notation and terminology of [LPP]we find (withoutany UMD assumption) that if X, Y are two 2-convex Banach lattices, then there is a naturalinclusion Rad(X ˆ⊗Y ) ⊂X(ℓ2)ˆ⊗Y + X ˆ⊗Y (ℓ2), so that if X, Y are both of finite cotype wehaveRad(X ˆ⊗Y ) ≈Rad(X)ˆ⊗Y + X ˆ⊗Rad(Y ).This is a slight refinement of some of the results of [LPP].Remark 4.4.

As in Remark 1.9, let us denote by H1(T; X) (resp. H1(TIN; X)) the subspaceof L1(T, m; X) (resp.

L1(TIN, mIN; X)) formed by the functions with Fourier transformsupported by the non-negative elements. We define similarly (for short) the space H∞(T; X)(resp.

H∞(TIN; X)). Using Remark 1.9 we obtain the same conclusion as Theorem 4.2whenever there is a function f in the interior of the unit ball of H1(T; X ⊗Γ Y ) (resp.H1(TIN; X ⊗Γ Y )) such that ˆf(3k) = ψk (resp.

ˆf(zk) = ψk) for all k = 1, ..., n. Of course thisremark applies in particular to Theorem 2.2. In the case of Theorem 2.2 this remark seemsuseful because it turns out the converse is true.

More precisely using a classical inequality(cf. [R] p.222) it can be proved that, in the situation of Theorem 2.2, for every element inthe unit ball of ℓ∞(S, ℓ2)ˆ⊗ℓ∞(T) with a finitely supported sequence of coefficients (Ak) inℓ∞(S)ˆ⊗ℓ∞(T) there is a function f in H∞(T; ℓ∞(S)ˆ⊗ℓ∞(T)) ⊂H1(T; ℓ∞(S)ˆ⊗ℓ∞(T)) withnorm less than an absolute constant C such thatˆf(3k) = Ak.28

We can treat similarly any element in the unit ball of ℓ∞(S)ˆ⊗ℓ∞(T, ℓ2), hence the sameconclusion holds for any element (with a finitely supported sequence of coefficients) in theunit ball of the spaceℓ∞(S, ℓ2)ˆ⊗ℓ∞(T) + ℓ∞(S)ˆ⊗ℓ∞(T, ℓ2).In other words, the point of the present remark is that it yields a characterization of thesequences (ψk) for which the conclusion of Theorem 2.2 holds.29

§ 5. More applications to completely bounded maps.As emphasized in [BP] (see also [P4]) the cb norm on B(Mn, B(H)) can be viewed asan example of the Γ norms discussed in section 4.

In particular we can obtain an analogueof Theorem 2.2 for c.b. maps.Theorem 5.1.

Consider Hilbert spaces H1 and H and completely bounded mapsu1, ..., uN : B(H1) →B(H).Assume thatR NPk=1zkukcbdµ(z) < 1. Then there is for some Hilbert space K a represen-tationπ : B(H1) →B(K)and operators Vk : H →K, Wk : H →K, V : H →K W : H →K such that∥V ∥≤1, ∥W∥≤1,NX1Vk∗Vk ≤1NX1Wk∗Wk ≤1and such that for all k = 1, ..., N∀x ∈B(H1) uk(x) = V ∗k π(x)W + V ∗π(x)Wk.Remark 5.2.

First observe that it is enough to find representations π′ : B(H1) →B(K′)π′′ : B(H1) →B(K′′) such that uk(.) = V ∗k π′(.

)W + V ∗π′′(. )Wk since we canreplace each of π′ and π′′ by π′ ⊕π′′.Remark 5.3.

Assume that we have a net (uαk)k≤N of N-tuples of maps from B(H1) intoB(H) such that for all x in B(H1)uαm(x) →uk(x) when α →∞and satisfying theconclusions of Theorem 5.1, i.e. such that there is a Hilbert space Kα, a representationπα : B(H1) →B(Kα) and operators V αk , V α, W αk , W αsuch that V α, W α, PkV α∗k V αkandPkW α∗k W αk are all of norm ≤1 and we have for all x in B(H1)uαk(x) = V α∗k πα(x)Wk + V α∗πα(x)W αk .Then {uk} satisfies the conclusions of Theorem 5.1.

Indeed, we can take for K an ultraprod-uct of the Hilbert spaces Kα, and similarly for π and for the operators V, W, Vk, Wk.Remark 5.4. Assume H1 and H both finite dimensional.

Then any operator u : B(H1) →B(H) can be identified with a linear operator ˜u : H1 ⊗H →(H1 ⊗H)∗defined by∀x1, y1 ∈H1∀x, y ∈H30

< ˜u(x1 ⊗x), y1 ⊗y >=< y, u(x1 ⊗y1)x > . (Note : We denote by y →y an anti isometry of H onto itself ; note that on the left sidewe have a bilinear pairing while the scalar product appearing on the right side is antilinearin the first variable).Consider a factorization of ˜u of the form˜u =i=nXi=1xi ⊗yiwith xi, yi ∈(H1 ⊗H)∗We define(5.1)δ1(nXi=1xi ⊗ei) = sup{|nXi=1< xi, hi ⊗h > |hi ∈H1,X∥hi∥2 ≤1, h ∈H, ∥h∥≤1}.We may identify an element xi in (H1 ⊗H)∗with a linear operator Vi : H →H1 by setting(5.2)∀k ∈H, ∀h ∈H< xi, k ⊗h >=< k, Vih >Then (5.1) becomes(5.3)δ1(nXi=1xi ⊗ei) = sup{(X∥Vih∥2)1/2|h ∈H∥h∥≤1}=(XV ∗i Vi)1/2.Let X = (H1 ⊗H)∗and let V be the linear span of the basis vectors of ℓ2.

Clearly theformula (5.3) defines a 2-convex norm on X ⊗ℓ2 (by density, say, of X ⊗V in X ⊗ℓ2).We set Y = X and we define δ2 = δ1 on Y ⊗ℓ2. Then we can define the norm Γassociated to δ1 and δ2 as in section 4, and also the norms ∆1 and ∆2 and the spacesX[ℓ2] ⊗Γ Y and X[ℓ2] ⊗Γ Y .By well known results on the factorization of c.b.

maps (cf. [Pa]) we have then∥u∥cb = Γ(˜u).Moreover, if u1, ..., uN are given cb maps from B(H1) into B(H), and if ˜u1, ..., ˜uN denotethe corresponding elements of X ⊗Y (with X = Y = (H ⊗H1)∗).

We claim that(5.4)NXk=1˜uk ⊗ekX[ℓ2]⊗ΓY< 131

iffthere are operators {W ki |k ≤N, i ≤n} and {Vi|i ≤n} such that ∀x ∈B(H1)uk(x) =nXi=1V ∗i xW ki andXiV ∗i Vi < 1Xi,kW ki∗W ki< 1.Indeed (5.4) holds iffwe can writeNXk=1˜uk ⊗ek =nXi=1ξi ⊗ηiwith ξi ∈X ⊗ℓN2 and ηi ∈Y such that∆1(Xξi ⊗ei) < 1andδ2(Xηi ⊗ei) < 1.Let ξi = Pk≤Nξki ⊗ek with ξki ∈X and let V ki and Wi be the operators associated to ξki andηi by the correspondence (5.2). We then obtain the above claim.This remark shows that in the case when both H1 and H are finite dimensional, Theorem5.1 can be viewed as a particular case of Theorem 4.1.Proof of Theorem 5.1.

We assume H finite dimensional until the last step of the proof. Thecase when H1 is also finite dimensional has been checked in the preceding remark.

AssumeH1 infinite dimensional assume that each u1, ...uN is weak*-continuous, i.e. continuousfrom σ(B(H1), B(H1)∗) into B(H).

We may use the fact that there is an increasing setB(Hα) ⊂B(H1) with dim Hα < ∞such that SαB(Hα) is weak*-dense in B(H1). Applyingthe first part of the proof to the restrictions uk|B(Hα) for each α and passing to the limit ina standard way (as in remark 5.3) we obtain Theorem 5.1 in that case also.Next when H1 is arbitrary and H finite dimensional we can involve the local reflexivityprinciple (cf.

e.g. [D]) to claim that there is a net (uαk)k≤N of maps which are weak*continuous from B(H1) into B(H), which tend pointwise to (uk)k≤N when α →∞andwhich satisfyZ Xzkuαkcb dµ(z) < 1.By remark 5.3 we obtain Theorem 5.1 in that case also.Finally we remove the assumption that H is finite dimensional.32

Let Hα be an increasing net of finite dimensional subspaces of H with S Hα = H.For each k and α letuαk(x) = PHαuk(x)|Hα ∈B(Hα).By the first part of the proof the conclusion of Theorem 5.1 holds for (uαk)k≤N for each α.Using an ultraproduct argument as in Remark 5.3 we conclude one more that u1, ..., uNsatisfy the conclusions of Theorem 5.1.We now give some consequences of Theorem 5.1. We denote again by ℓn2 the ndimensional Hilbert space equipped with its canonical basis (ei)i≤n.

We will identify Mnwith B(ℓn2) and eij with ei ⊗ej.We will need the following notation.Let H1, H2 be two Hilbert spaces and let S1 ⊂B(H1), S2 ⊂B(H2) be closed subspaces(“operator spaces”). We will denote by S1 ⊗min S2 the completion of S1 ⊗S2 equipped withthe norm induced by B(H1 ⊗2 H2), where H1 ⊗2 H2 is the Hilbertian tensor product.

Thespace S1 ⊗min S2 is called the minimal (or the spatial) tensor product of S1 and S2. Inparticular we have obviously Mn(B(H)) = Mn ⊗min B(H).We will denote by BRn (resp.BCn) the subspace of Mn ⊗min Mn formed by all elements ofthe formy1 =Xieii ⊗Xjxijeij)(resp.y2 =Xjejj ⊗Xixijeij.Note that ∥y1∥= supj(Pi|xij|2)1/2 and ∥y2∥= supi(Pj|xij|2)1/2 so that BRn (resp.

BCn)is naturally isometric with the space of matrices with “bounded rows” (resp. “boundedcolumns”), which explains our notation.Let J1 : Mn →BRn (resp.

J2 : Mn →BCn) be the map defined byJ1(x) =Xieii ⊗Xjxijeij(resp.J2(x) =Xjejj ⊗Xixijeij). )Obviously we have ∥J1∥≤1 and ∥J2∥≤1.

Moreover a simple verification shows that(5.5)∥J1∥cb ≤1 and ∥J2∥cb ≤1.33

Let us denote here G = Tn2 and let µ be the normalized Haar measure on G. By a simplecomputation one can chech that for any Hilbert space H and for any xij in B(H) we have(5.6)′Xieii ⊗Xjeij ⊗xijBRn⊗minB(H)= supiXjeij ⊗xijMn(B(H))= supi(Xjxijx∗ij)1/2B(H)and similarly(5.6)′′Xjejj ⊗Xieij ⊗xijBCn⊗minB(H)= supj(Xix∗ijxij)1/2B(H).Observe that the preceding expressions do not change if we replace (xij) by (zijxij) with|zij| = 1, i.e. with z = (zij) ∈G.

Hence if we denote by Tz : Mn →Mn the Schur multiplierdefined byTz((aij)) = (aijzij),we find using (5.5) and this observation that for all z = (zij) in G we have(5.7)∥J1Tz∥cb ≤1∥J2Tz∥≤1.We can now stateCorollary 5.5. Let H be a Hilbert space.

Consider an operator u : Mn →B(H). Letuz = uTz : Mn →B(H).

( i) AssumeR∥uz∥cb dµ(z) < 1. Then there are operators a1 : BRn →B(H) anda2 : BCn →B(H) such that(5.8)∥a1∥cb ≤1∥a2∥cb ≤1 and u = a1J1 + a2J2.

( ii) Conversely if (5.8) holds we have(5.9)Z∥uz∥cb dµ(z) ≤supz∈G∥uz∥cb ≤2.Proof. Note that (5.9) is an obvious consequence of (5.7) so it suffices to prove the firstpart.

Let uij : Mn →B(H) be defined by uij(x) = xiju(eij) so that uz = Pijzijuij. By34

Theorem 5.1 we can find a Hilbert space K operators V ij, W ij, V, W from H into K and arepresentation π : Mn →B(K) such that for all x in Mn(5.10)uij(x) = V ij∗π(x)W + V ∗π(x)W ijand∥V ∥≤1, ∥W∥≤1,XijV ij∗V ij≤1,XijW ij∗W ij≤1.Let Tij = u(eij). We deduce from (5.10)(5.11)Tij = V ij∗π(eij)W + V ∗π(eij)W ij.Let a1 : BRn →B(H) and a2 : BCn →B(H) be defined bya1(eii ⊗eij) = V ∗π(eij)W ijanda2(ejj ⊗eij) = V ij∗π(eij)W.Clearly by (5.11) we have u = a1J1 + a2J2.

We claim that ∥a1∥cb ≤1 and ∥a2∥cb ≤1. Tocheck this we will use the well known inequality(5.12)Xa∗kbk ≤(Xa∗kak)1/2(Xb∗kbk)1/2)valid for ak, bk ∈B(H).

For any ξij in B(K1)(K1 an arbitrary Hilbert space) we haveXa1(eii ⊗eij) ⊗ξij =XV ∗π(eij)W ij ⊗ξij≤(XW ij∗W ij)1/2(X(V ∗π(eij) ⊗ξij)(V ∗π(eij) ⊗ξij)∗)1/2≤∥V ∥(Xijπ(eij)π(eij)∗⊗ξijξ∗ij)1/2≤(Xjπ(ejj) ⊗Xiξijξ∗ij)1/2≤supj(Xiξijξ∗ij)1/2=Xijeii ⊗eij ⊗ξijBRn⊗minB(K1).35

Taking B(K1) = Mn with n ≥1 arbitrary, we obtain (recall (5.6))∥a1∥cb ≤1.Similarly we have ∥a2∥cb ≤1. This concludes the proof.We now turn to a generalized version of Schur multipliers.We consider an operator T : Mn(B(H1)) →Mn(B(H)) (where H, H1 are Hilbert spaces) ofthe special form(5.13)T((xij)) = (Tij(xij))where Tij : B(H1) →B(H) are operators.Remark 5.6.

Assume ∥T∥cb ≤1. Then there is a Hilbert space K, a representationπ : B(H1) →B(K) and operators xi, yj : H →K such that for all i, j(5.14)∀x ∈B(H1)Tij(x) = x∗i π(x)yj and ∥xi∥≤1, ∥yj∥≤1.Conversely, it is clear that (5.14) implies ∥T∥cb ≤1.This statement generalizes Proposition 1.1 to the present setting.Such a statement is a simple consequence of the factorization theorem of c.b.

maps (cf. [Pa]) and of the particular form (5.13) of the map T.Corollary 5.7.

Let T : Mn(B(H1)) →Mn(B(H)) be an operator of the form (5.13)(generalized Schur multiplier). As above, let G = Tn2 and let µ be the normalized Haarmeasure on G. Let Tz : Mn(B(H1)) →Mn(B(H)) be the operator defined by∀z = (zij) ∈GTz((xij)) = (zijTij(xij)).AssumeR∥Tz∥cb dµ(z) < 1.

Then there is a decomposition T = α1 + α2 where α1 and α2are each of the form (5.13) and moreover there are operators˜α1 : BRn ⊗min B(H1) →Mn(B(H))and˜α2 : BCn ⊗min B(H1) →Mn(B(H))satisfying∥˜α1∥cb ≤1, ∥˜α2∥cb ≤1,36

and such that(5.15)α1 = ˜α1(J1 ⊗IB(H1))α2 = ˜α2(J2 ⊗IB(H1)).Conversely, if such a decomposition holds then necessarilyZ∥Tz∥cb dµ(z) ≤supz∈G∥Tz∥cb ≤2.Proof. Let us define again ˜Tij : Mn(B(H1)) →Mn(B(H)) by the identityTz =Xijzij ˜Tij.Then, by Theorem 5.1 there are a Hilbert space ˜K and a representation ˜π : Mn(B(H1)) →B( ˜K) together with operators V, W, V ij, W ij from ℓn2(H) into ˜K such that(5.16)∀ξ ∈Mn(B(H1))˜Tij(ξ) = V ij∗˜π(ξ)W + V ∗˜π(ξ)W ij∗and such that(5.17)∥V ∥≤1, ∥W∥≤1,XijV ij∗V ij≤1,XijW ij∗W ij≤1.By standard arguments, we can assume w.l.o.g.

that ˜K = ℓn2(K) for some Hilbert spaceK and that ˜π : Mn(B(H1)) →B( ˜K) = Mn(B(K)) is of the form ˜π = IMn ⊗π for somerepresentation π : B(H1) →B(K). Once we identify ˜K with ℓn2(K) we may identify each ofV, W, V ij, W ij with an n × n matrix of operators from H into K.Thus we identify V with (V (k, ℓ))k,ℓ≤n V ij with (V ij(k, ℓ))k,ℓ≤n, and so on.Let now x be arbitrary in B(H1).

We have by (5.16)Tij(x) = ( ˜Tij(eij ⊗x))ij= [V ij∗(eij ⊗π(x))W + V ∗(eij ⊗π(x))W ij]ijhence(5.18)Tij(x) = V ij(i, i)∗π(x)W(j, j) + V (i, i)∗π(x)W ij(j, j).By (5.17) we have∥W(j, j)∥≤1, ∥V (i, i)∥≤1.37

Moreover, we claim that (5.17) implies(5.19)supiXjV ij(i, i)∗V ij(i, i)≤1 and supjXiW ij(j, j)∗W ij(j, j) ≤1.Indeed, (5.17) implies that for each ℓ= 1, 2, ..., nXijV ij(ℓ, ℓ)∗V ij(ℓ, ℓ)≤1hence a fortiori for each i (taking ℓ= i)XjV ij(i, i)∗V ij(i, i)≤1.The other estimate is proved similarly, hence the above claim.Finally, letα1((xij)) = (α1ij(xij)) and α2((xij)) = (α2ij(xij))where we define for all x in B(H1)α1ij(x) = V (i, i)∗π(x)W ij(j, j) and α2ij(x) = V ij(i, i)∗π(x)W(j, j).Clearly we can write (5.15) for some uniquely defined operators ˜α1 and ˜α2 as in Corollary5.7. We have∀x ∈B(H1)˜α1(ejj ⊗eij ⊗x) = eij ⊗α1ij(x)and˜α2(eii ⊗eij ⊗x) = eij ⊗α2ij(x).Finally, it remains to check that ˜α1 and ˜α2 acting on the spaces indicated in Corollary 5.7are of c.b.

norm at most 1.Let y1 ∈BRn⊗minB(H1) be of norm ≤1. Let y1 = Pieii⊗Pjeij⊗xij with xij ∈B(H1).We have˜α1(y1) =Xijeij ⊗V (i, i)∗π(xij)W ij(j, j).Observe that for all yij in B(H)Xeij ⊗yijMn(B(H)) =Xeij ⊗eij ⊗yijMn(Mn(B(H))38

Now let yij = V (i, i)∗π(xij)W ij(j, j).We haveXijeij ⊗eij ⊗yij = (Xijeii ⊗eij ⊗V (i, i)∗π(xij)). (Xijeij ⊗ejj ⊗W ij(j, j)).Hence we have∥˜α1(y1)∥≤Xijeii ⊗eij ⊗V (i, i)∗π(xij).Xijeij ⊗ejj ⊗W ij(j, j)hence by (5.6)’, (5.6)” and (5.19).∥˜α1(y1)∥≤supi(∥V (i, i)∥(Xjxijx∗ij)1/2) ≤∥y1∥BRn⊗minB(H)This shows that ∥˜α1∥cb ≤1.

Similarly we have ∥˜α2∥cb ≤1. This concludes the proof.Final Remark : Recently, C.Le Merdy has shown that Theorem 5.1 and corollary 5.7remain valid if B(H1) is replaced by an arbitrary C∗-algebra A ⊂B(H1).

Indeed, he hasproved (cf. [LeM]) that any bounded analytic function with values in the space CB(A, B(H))(i.e.

the space of all c.b. maps from A into B(H)) can be extended to a bounded analyticfunction (with the same H∞norm) with values in the space CB(B(H1), B(H)).

In otherwords, there is a way to extend c. b. maps from A into B(H) to c. b. maps defined on thewhole of B(H1) which preserves analyticity. This extends a result due to Haagerup and theauthor corresponding to the particular case when H is of dimension 1.

Using Le Merdy’sresult and Remark 4.4 it is rather easy to adapt the proof of Theorem 5.1 (or corollary 5.7)with B(H1) replaced by any C∗-subalgebra A ⊂B(H1).39

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