Fourier analysis, Schur multipliers on S p and non-commutative Λ(p)-sets

Transcription

1 Fourier analysis, Schur multiliers on S and non-commutative Λ-sets Asma Harcharras Abstract This work deals with various questions concerning Fourier multiliers on L, Schur multiliers on the Schatten class S as well as their comletely bounded versions when L and S are viewed as oerator saces. We use for this aim subsets of Z enoying the non-commutative Λ-roerty which is a new analytic roerty much stronger than the classical Λ-roerty. We start by studying the notion of non-commutative Λ-sets in the general case of an arbitrary discrete grou before turning to the grou Z. AMS classification 1991: rimary 43A, 43A46, 46L50, secondary 46B70, 47B10, 47D15. 1

2 Contents 0 Introduction, background and notation Comlex interolation Oerator saces Non-commutative Khintchine inequalities An exlicit descrition of the saces S,unc and S,unc S Additional non standard notations Peller s theorem Some suitable oerator norms inequalities Fourier multiliers Schur multiliers Non-commutative Λ-sets in discrete grous 17 Non-commutative Λ-sets in Z 30 3 Alications to Fourier multiliers 34 4 σ-sets and σ cb -sets 36 5 Alications to Schur multiliers 41 6 Aendix 44

3 0 Introduction, background and notation ML stands for the algebra of Fourier multiliers on the sace L, and M cb L for the algebra of Fourier multiliers which are comletely bounded on L when the latter is endowed with its natural oerator sace structure. MS denotes the algebra of Schur multiliers on the Schatten class S, and M cb S the algebra of all Schur multiliers which are comletely bounded on S when the latter is equied with its natural oerator sace structure. Our first motivation was to show that the following contractive inclusion mas M cb L q M cb L, MS q MS, ML, ML M cb L, MS, MS θ θ M cb S where in the three first inclusions is an even integer and < < q while in the two last ones 0 < θ < 1 is arbitrary and = are all strict. The reader should note that θ the embeddings in which we are interested as well as the isomorhisms we consider in this work are the natural ones meaning those which send a given element simly to itself. For this aim we introduce and study a non-commutative version of the usual Λ-sets. The idea behind all the roofs is the existence for each even integer < < of a non-commutative Λ-set which is not a Λq-set for any q >. For the rest of this section, we recall all the facts we need along with the notations we use in the sequel. Section 1 is devoted to the study of the non-commutative Λ-roerty in an arbitrary discrete grou G. This is a new analytic roerty more restrictive in general than the classical Λ-roerty. We start the section by recalling the definition of Λ-sets and we oint out their relationshi with the set M L τ 0 of all Fourier multiliers on L τ 0. Here the sace L τ 0 denotes the non-commutative L -sace associated to the discrete grou G equied with its usual trace τ 0. Then we introduce the non-commutative Λ- sets. We oint out their relationshi with the set M cb L τ 0 of all comletely bounded Fourier multiliers on L τ 0 when the latter is endowed with its natural oerator sace structure. This ustifies the terminology Λ cb -sets we use for non-commutative Λ- sets. The links between Λ-sets and the algebra M L τ 0 on one hand and between Λ cb -sets and M cb L τ 0 on the other are roved by using the non-commutative version of Khintchine inequalities roved in [6] see also [7]. Then we consider for integers two combinatorial roerties defined on subsets of G namely the B-roerty and the Z-roerty. We show that the B-roerty ensures the Z-roerty and that the Z-roerty ensures the Λ cb -roerty; the latter result is the crucial oint of this work. In Section, we consider the Λ cb -roerty in the articular case of the grou Z. We rove that this roerty is very different from the usual Λ-roerty. More recisely, we rove that there exists a set which is Λ for each < < but not Λ cb for any < <. Then we show that for each even integer > there exists a Λ cb -set which is not a Λq-set for any q > ; this kind of articular sets will lay the key rôle in the roofs of the results announced in the following sections. 3

4 In section 3, we focus on Fourier multiliers. We rove that given < an even integer, M cb L cannot embed continuously into ML q for any < q. Recall that for = ML and 0 < θ < 1, the embedding of, ML into ML is strict see [45], θ see also [40]. Then since as we will recall the embedding of M cb L into ML is strict for any < <, it is natural to wonder whether the embedding of ML, ML into M cb L is again strict. We rove that this is still the case. More recisely, we show that M cb L does not embed continuously into the interolated sace ML, ML for any 0 < θ < 1. In Section 4, we introduce and study the so-called σ-sets and σ cb -sets. These are subsets of N N laying for MS and M cb S a rôle analogous to the one layed by Λ-sets and Λ cb -sets for ML and M cb L resectively. We will see that from any given Λ cb -set, we can obtain a σ cb -set and thus we get for even integers secial σ cb -sets. Indeed, we rove that for any even integer >, there is a σ cb -set A N N which is not a σq-set for any q >. Section 5 is devoted to Schur multiliers. For each even integer < <, we rove the existence of an idemotent Schur multilier which is comletely bounded on S but not bounded on S q for any < q. In fact, our idemotent Schur multilier is not even bounded on the subsace of S q formed of all Hankelian oerators denoted S q in the sequel. This answers a question raised by J. Erdos. Therefore, the embeddings MS q MS and M cb S q M cb S are strict whenever < q and is an even integer. On the other hand, we show that for each given < <, the set M cb S does not embed continuously into the interolated sace MS, MS for any 0 < θ < 1. This answers a question raised by V. Peller. We will take the oortunity in this section to establish links between Fourier and Schur multiliers as follows. Let MH be the algebra of Fourier multiliers on the Hardy sace H, M cb H be the algebra of comletely bounded Fourier multiliers on H, MS be the algebra of Schur multiliers on S and M cb S be the algebra of comletely bounded Schur multiliers on S. The saces H and S are viewed as oerator subsaces of L and S resectively. We show that MH can be inected continuously into MS in the same way M cb H is inected into M cb S. For this urose, we are led to characterize the multiliers of MS and M cb S our characterizations are easy consequences of [9], [30]. Section 6 is included for the sake of comleteness. Using robabilistic ideas, we exhibit a very large Z-set roughly the largest ossible one which enoys some additional roerties. On the other hand, we introduce on the subsets of N N some simle combinatorial roerties ensuring the σ4 cb -one, which we call roerty C and roerty R. Then by using similar robabilistic ideas, we exhibit large sets satisfying one of these combinatorial roerties. Acknowledgments. This work owes a great deal to the author s advisor, Professor Gilles Pisier. His recious hel and availability made it ossible thus we would like to exress our gratitude to him. Some results were achieved during a stay at Texas A&M University which we thank for its hositality and financial suort. θ θ θ θ 4

5 We now review the standard notation that we use. Let E and F be two Banach saces. By E F, we denote the algebraic tensor roduct of E and F. By BE, F we denote the set of all bounded oerators from E to F. BE, E is simly denoted BE. B E stands for the oen unit ball of E. id E denotes the identity ma on E. If E i, F i are Banach saces and u i is in BE i, F i for i = 0, 1 then u 0 u 1 denotes the oerator which carries x y in E 0 E 1 to u 0 x u 1 y in F 0 F 1 extended linearly. A contractive ma u : E F is said to be µ-surective if the set uµb E contains B F. 1-surective mas are called metric surections. For 1, Ω, ν a measurable sace and E an arbitrary Banach sace, we let L Ω, dν, E be the set of all E-valued functions f on Ω which are Bochner measurable and such that f L Ω,dν,E := Ω ft dν < When Ω is the torus T and ν is the normalized Lebesgue measure, the sace L T, dν, E is simly denoted by L E and we let H E be the E-valued Hardy sace namely this is the set of all f in L E such that the Fourier coefficients fn = 0 for all integers n < 0. L C and H C are simly denoted L and H resectively. More generally, let M be a von Neumann algebra given with a normal, faithful and semifinite trace τ M. For 1 <, L τ M denotes the non-commutative L -sace associated to M equied with τ M. By definition, this is the Banach sace obtained from the sace of all x in M satisfying x L τ M := τ M x x < after comletion with resect to the norm. L τ M cf. [16], [8], [38]. By convention, L τ M denotes M. The noncommutative L -sace associated to BH where H denotes a searable Hilbert sace, equied with its usual trace is nothing but the -Schatten class on H. It will be denoted by S H when 1 <. S H stands for the set of all comact oerators on H. In the articular case H = l res. l n the n-dimensional Hilbert sace, the usual trace on Bl res. M n := Bl n is denoted tr res. tr n and the sace S H is simly denoted S res. Sn for each 1. If τ M and τ N are normal, faithful and semi-finite traces given on the von Neumann algebras M and N resectively, then we let τ M τ N denote the trace on the von Neumann algebra generated by M N defined as follows. τ M τ N x y := τ M xτ N y, x M, y N. τ M τ N is still normal, faithful and semi-finite thus we can consider unambiguously the sace L τ M τ N. Given a discrete grou G, denotes the left regular reresentation of G into B l G, L τ 0 denotes the non-commutative L -sace associated to the von Neumann algebra generated by G with resect to its usual trace denoted τ 0, and L τ denotes the noncommutative L -sace associated to the von Neumann algebra generated by G Bl with resect to the trace τ = τ 0 tr. Given M and τ M as above, the saces L τ M tr and L 1 τ M tr form a comatible coule of comlex interolation for which we have isometrically cf. [0] 1 < <, L τ M tr = L τ M tr, L 1 τ M tr This allows us to see the Banach saces L τ M in the recent view oint of oerator saces in a natural way cf. [33], [34]. 5

6 0.1 Comlex interolation Let E 0, E 1 be a air of comatible Banach saces i.e. E 0 and E 1 are both continuously inected into the same toological sace. Let := z C 0 Rez 1 := z C Rez = for = 0, 1. Then let GE 0, E 1 be the set of all functions f of the form f = where the x k s are in E 0 E 1, the functions f k : C are continuous on analytic on its interior and vanishing at infinity. Denote by FE 0, E 1 the comletion of GE 0, E 1 for the norm f := max fz fz su, su z 0 E0 z 1 E1 For 0 < θ < 1, consider the subset N θ E 0, E 1 of GE 0, E 1 of all functions which vanish on θ and let S θ E 0, E 1 be its closure in FE 0, E 1. By definition, the intermediate sace E θ obtained by comlex interolation between E 0 and E 1 corresonding to the value θ is the Banach sace FE 0, E 1 /S θ E 0, E 1 equied with the quotient norm denoted. θ. We refer the reader to [39] to make sure that the definition we chose for the comlex interolation coincides with the one given in []. finite f k x k Lemma 0.1 Let E 0, E 1 and F 0, F 1 be two comatible coules of interolation such that E 0 E 1 is dense in both E 0 and E 1. Then the sace BE 0, F 0, BE 1, F 1 embeds contractively into BE θ, F θ for each 0 < θ < 1. Proof: Let E be the comletion of E 0 E 1 for the norm x xe = max E0, xe1 and F be any Banach sace containing continuously F 0 and F 1. The assumtion on the air E 0, E 1 ermits to inect continuously both BE 0, F 0 and BE 1, F 1 into BE, F. Thus they form a comatible coule of interolation. By density of BE 0, F 0 BE 1, F 1 in BE 0, F 0, BE 1, F 1 and E 0 E 1 in E θ, we are reduced to show that Txθ Tθxθ for each T in BE 0, F 0 BE 1, F 1 and each x in E 0 E 1. To check this, let ϕ be in G BE 0, F 0, BE 1, F 1, f in GE 0, E 1 such that ϕθ = T and fθ = x and consider the function g which takes z in to ϕz fz in F 0 F 1. Clearly g belongs to GF 0, F 1 with gθ = Tx. Moreover, its norm satisfies ϕz g = max su ϕz fz max su fz =0,1 z F =0,1 z BE,F E g ϕz fz max su max =0,1 z BE su = ϕ f,f =0,1 z E This gives the required inequality after taking the infimum over all such ϕ s and f s. θ θ 6

7 0. Oerator saces Concretely, by an oerator sace, we mean a closed subsace of BH for some Hilbert sace H. Such an obect has natural norms. n on M n E the set of n n matrices with entries in E. Indeed, M n E can be viewed as a subsace of Bl n H via the natural identification between M n BH and Bl n H. This sequence of norms satisfy Ruan s axioms, that is a, b M n, x M n E we have a x bn amn xnbmn x M n E, y M m E we have x x yn+m = max n, ym Here the norm on M n is the oerator norm, the denotes the direct sum of matrix and the dot denotes the matrix roduct following the usual rules of calculation. In the oerator settings, a ma u : E F is said to be c.b. short for comletely bounded if the mas u n : M n E x i i, M n F ux i i, are uniformly bounded. We let CBE, F stand for the sace of all c.b. mas endowed with the norm ucb = su u n. n CBE will stand for CBE, E. An oerator u is said to be a comlete contraction res. isometry if each ma u n is contractive res. isometric. Z. Ruan gave an abstract characterization of an oerator sace as a Banach sace given with a sequence of norms on the M n E s which satisfy Ruan s axioms see [36]. This abstract characterization allows to define for oerator saces the notion of duality, comlex interolation... The standard dual of an oerator sace E is the usual Banach sace E with the norms corresonding to the isometric identifications of M n E with CBE, M n as in [4]and [17]. The comlex interolated sace between two oerator saces E 0 and E 1 comatible as Banach saces is the usual Banach sace E θ with the norms corresonding to the isometric identifications M n E θ := M n E 0, M n E 1 see [33]. When E and F are two oerator saces, CBE, F is an oerator sace for the structure corresonding to the isometric identifications M n CBE, F := CB E, M n F. Note that the min. short for minimal tensor roduct is a very useful tool to describe entirely the oerator sace structure of an oerator sace as well as the c.b. mas between oerator saces. Let E BH be a concrete oerator sace. By S min E, we mean the comletion of S E for the norm induced by Bl H. Then the oerator sace structure of the interolated sace E θ and the one of the oerator sace dual E are entirely described by the following isometric relations: S min E CBE, S S min E θ = S min E 0, S min E 1 θ A ma u : E F is c.b. if and only if id S u extends to a bounded oerator from S min E into S min F and we have u cb = id S u : S min E S min F. 7 θ

8 Lemma 0. Let E 0, E 1 and F 0, F 1 be two comatible coules of interolation. Assume that E 0 E 1 is dense in both E 0 and E 1. Then CBE 0, F 0, CBE 1, F 1 embeds comletely contractively into CBE θ, F θ for each 0 < θ < 1. Proof: For arbitrary oerator saces E and F we may view CBE, F as a subsace of BS min E, S min F via the isometric embedding which carries an oerator T in CBE, F to the oerator id S T in BS min E, S min F. Now let E 0, E 1, F 0 and F 1 be as above. Lemma 0.1 alied to the new airs S min E 0, S min E 1 and S min F 0, S min F 1 imlies that for each real number 0 < θ < 1, the sace B S min E 0, S min F 0, B S min E 1, S min F 1 embeds contractively into the θ sace B S min E θ, S min F θ. This imlies that CBE 0, F 0, CBE 1, F 1 embeds contractively into CBE θ, F θ. Actually, the embedding is comletely contractive. Indeed, this gives for each integer n 1 M n CBE0, F 0, CBE 1, F 1 = M θ n CBE0, F 0, M n CBE1, F 1 = θ CB E 0, M n F 0, CB E 1, M n F 1 CB E θ, M n F 0, M n F 1 = CB E θ θ θ, M n F θ Thus the embedding M n CBE0, F 0, CBE 1, F 1 M θ n CB E θ, F θ is contractive. G. Pisier roved in [34] that in fact the theory of oerator saces can be develoed equivalently using other sequences of norms on the M n E s. Indeed, let 1 be a fixed number, let E be an oerator sace and let E be its dual oerator sace. For an integer n 1, we let Sn[E] denote the sace M n E but equied with the norm below Sn [E] := Sn [E], S1 n [E] where Sn [E] denotes M ne for convenience only, Sn 1[E] is the sace M ne viewed as a subsace of M n E and θ = 1. Note that S n [E] embeds isometrically into S n+1 [E] thus we set S [E] for the comletion of n 1S n [E]. a.x.b Proosition 0.3 [34] For all x in M n E, we have x = su MnE Sn[E] where the suremum runs over all a, b in the unit ball of Sn. Therefore an oerator u : E F is c.b. if and only if the mas u n : Sn [E] S n [F] are uniformly bounded in which case we have u u cb = su n : Sn[E] Sn[F]. n 1 Now let us go back to the case of non-commutative L -saces. If M is a von Neumann algebra given with a normal, faithful and semi-finite trace τ M then since L τ M is a C - algebra, it has a natural oerator sace structure given by any concrete realization as a C -subalgebra of some BH. Since L 1 τ M coincides with the redual of L τ M, it aears also as an oerator sace in a natural way. Indeed, it is a subsace of the standard dual of L τ M. Hence the saces L τ M are also canonically endowed with an oerator sace 8 θ θ θ

9 structure, the one obtained by comlex interolation in the oerator saces category. Alying Pro. 0.3 we get a nice and a simle characterization of the c.b. mas between these saces since for each integer n 1, we have the following natural identifications S n [ L τ M ] [ = L τ M tr n, Sn 1 L 1 τ M ] = L 1 τ M tr n These imly that we have isometrically Sn[ L τ M ] [ = Sn L τ M ] [, Sn 1 L 1 τ M ] θ = L τ M tr n, L 1 τ M tr n θ = L τ M tr n Therefore a density argument yields S [ L τ M ] = L τ M tr isometrically. Thus Pro. 0.3 imlies Proosition 0.4 Let 1 <, L τ M and L τ N be two non-commutative L -saces and E L τ M, F L τ N arbitrary oerator subsaces. Then an oerator u : E F is c.b. if and only if the oerator u id S : E S F S which takes x y to ux y where x E and y S, extends to a bounded oerator from E S L τ M tr into F S L τ N tr. Moreover we have ucb = u id S : E S L τ M tr F S L τ N tr. 0.3 Non-commutative Khintchine inequalities Let ε n : 1, 1 N 1, 1 be the n-th coordinate roection, ν the uniform robability measure on 1, 1 N and 1 < an arbitrary real number. In the commutative case, the classical Khintchine inequalities say that there exists a constant k > 0 deending only on such that for all integers n 1 and all scalars x 1, x,...x n we have n x when 1 L n ε x k 1,1 N,ν =1 n L n ε x k 1,1 N,ν =1 =1 =1 x 1 when < 0.1 See e.g. [3] for the roof. Later on these inequalities were generalized to the noncommutative case by F. Lust Piquard for 1 < < cf. [6] and by F. Lust Piquard and G. Pisier for = 1 cf. [7] as follows. Let M be a von Neumann algebra given with a normal, faithful and semi-finite trace τ M. For each 1 <, there exists a ositive constant K deending only on the air M, τ L τm and such that for all n 1 in N M and all x 1, x,...x n in L τ M we have in the case of 1 n L n 1 ε x K inf L τm y y n 1 zz =1 1,1 N,ν,L τ M =1 L τ M + =1 L τ M 0. where the infimum runs over all decomositions of the x s in L τ M as y + z, while n L n ε x K max L τm x x n 1 x =1 1,1 N,ν,L τ M =1 L τ M, x =1 L τ M 0.3 in the case of <. In the articular case of S the constant K S will be denoted K for simlicity. 9

10 0.4 An exlicit descrition of the saces S,unc and S,unc S An oerator in Bl will be frequently identified with its corresonding matrix relatively to the canonical basis of l. Let Ω 0 be the set 1, 1 N N, ν be the uniform robability measure on Ω 0 and let ε i : Ω 0 1, 1 be the i, -th coordinate roection. For 1, S,unc denotes the sace of all oerators x = x i i, in S such that the oerators ε i x i belong to i, S for almost all choices of signs ε i on N N, equied i, with the norm below x := S,unc ε i x i i, L Ω 0,ν,S We mean by S S the set of all matrices x = x i i, with entries x i in S and which are viewed as oerators on l l in the -Schatten class on the Hilbert sace l l, equied with the inherited norm Note that S S is exactly the -Schatten class on l l via the identification mentionned above. Then similarly, we let S,unc S be the set of all oerators x = x i i, in S S with entries in S such that the oerators εi x i i, are in S S for almost all choices of signs ε i on N N, equied with the i, norm below x := S,uncS ε i x i i, L Ω 0,ν,S S The next result essentially goes back to F. Lust Piquard [6]. Lemma 0.5 There is an exlicit descrition of the sace S,unc res. S,unc S for each 1 < as follows. For all x = x i i, in S,unc res. S,unc S we have when < res. x x S,unc S,unc S while for 1 res. x x S,unc = inf S,unc S = max i = max = inf i i x i, 1 x ix i y i i x i 1, x i x S i + i i z i yiy i 1 + z i z S i i i 1 S 1 S where the infimum runs over all ossible decomositions of x as a sum of y = y i i, and z = z i i, both in S res. S S. Proof: We rove the lemma for S,unc S when < only, the other cases are quite similar and left to the reader. We start by recalling that for each and each x = x i i, in S S we have x x S S max ix i 1, x i x S i i 10 i S

11 Indeed, this holds for = and =. Then using comlex interolation, it is also satisfied for all < <. Now let < be fixed. When x = x i belongs to i, S,unc S, the matrices ε i x i satisfy 0.4 for almost all the choices of signs i, εi i, on Ω 0 since the matrices ε i x i belong to i, S S almost surely. Thus after integrating over all these choices of signs, we get for each x = x i i, in S,unc S x x S,unc S max i x i i 1, x i x S i i 1 S The converse inequality is obtaind with the hel of the inequality 0.3. Indeed, we have x = S,unc S Ω 0 i, N N ε i x i e i S S dν K max x i e i x i e i S i, S, 1 x i e i x i e i i, 1 = K max x i x 1 i e x i x S S, i e ii i, = K max x i x i i S e S, i x K max x S,unc S i x i i Additional non standard notations i, S S S S 1 x i x S i e ii S 1, x i x S i i 1 1 S For the rest of this aer, we will use frequently the following definitions and notations. A given matrix x = x kl k,l where the entries x kl belong to some fixed set is said to be Hankelian if it satisfies x kl = x k l whenever k + l = k + l. For 1 <, S res. S S will stand for the subsace of S res. S S formed of all Hankelian matrices x = x kl k,l in S res. S S. For a set Λ, 1l Λ stands for its indicator function and Λ stands for the cardinality of Λ. If Λ is a subset of a discrete grou G and if F denotes either L τ 0 or L τ for some 1 then we let F Λ := f F ft = 0, Recall that for f in L τ 0 res. L τ the Fourier coefficient ft is defined as follows ft = τ 0 [ t 1 f ] res. ft = τ 0 id S [ t 1 id l f ] We denote simly F G by F and when G = Z and Λ = N, L N will be still denoted by H. Similarly when F is a class of matrices and A is a subset of N N we let F A := x = x kl F x k,l kl = 0, 11

12 F N N is denoted simly by F. Moreover when F is a Banach or an oerator sace the sets F Λ and F A are automatically viewed as Banach or oerator subsaces of F. When Λ N, Λ := k, l N N k + l Λ and any subset of N N which can be written as Λ for some set Λ N is called a Hankelian set. Then given a ma ϕ : Λ C we let ϕ : Λ C k, l ϕk + l. For an analytic function f on T we let for all z in T f 0 z := f0 while for all integers n 1 f n z := n 1 k= n 1 fkzk Similarly given an matrix x we let x 0 := x 00 while for all integers n 1 we let x n := x1l k,l n 1 k+l< n Schur roduct. 0.6 Peller s theorem The aim of Peller s theorem is to realize S and more generally S S as a sace of functions on the torus T which we will describe here for 1 < < only. Consider the Banach saces Besov saces A := f : T C analytic fa < A S := g : T S analytic gas < where f A := n fn n=0 L ga := n gn S n=0 L S Theorem 0.6 The following mas are well defined, bounded and biective. [ ] [ A S A S S S and f fk + l g ĝk + l k,l 0 k,l 0 ] In other words, as Banach saces, A is isomorhic to S and A S is isomorhic to S S in a canonical way. In the case of S we refer the reader to Section of the aer [9], the norm of A as described above is given exlicitly in age 450 while we refer to Section 3 of [30] for the case of S S, the norm of A S described above is then imlicit. Therefore we have x S S, x n n = n 1 k= n 1 x 0k z k L xs = n=0 x 0k z k 1 L k= n 1 n n 1 1

13 and similarly for all x in S S, we have x n S S n = n 1 k= n 1 x 0k z k L S xs S = n=0 n n 1 k= n 1 x 0k z k L S x 0k z k := 0 when k = 1. These descritions rovide S and S S with very useful equivalent norms as follows. Corollary 0.7 i For each fixed 1 < <, the following are equivalent norms on S x S = x n n=0 S, x S. ii For each fixed 1 < <, the following are equivalent norms on the sace S S x S S = x n n=0 S S, x S S. 0.7 Some suitable oerator norms inequalities Proosition 0.8 Consider 1 q, α, β > 1 such that = 1, y a ositive α β oerator in S qα and x n n a finite sequence of oerators each in Sqβ, then we have n x n yx n y max x n x n S q S qα n S qβ, n x n x n This roosition goes back to [6] when x n is a family of self-adoint oerators. The n general case for which the roof uses basically the three line lemma can be found in [35]. The next corollary follows easily by a reiteration argument. Corollary 0.9 Let 1 q, r 1 and for each 1 r, let I be a finite set of r 1 indexes, α > 1 with = 1 and x n α be a family of oerators each in S n qα. I Then we have n I 1 r =1 x n r r...x n x 1 n 1 x 1 n 1 x n...x r 0.8 Fourier multiliers n r S q r =1 max x n x S n, qα n I S qβ n I x n x n S qα A scalar valued ma ϕ on Λ G is said to be a Fourier multilier on L Λ τ 0 if the associated oerator M ϕ : san t, t Λ san t, t Λ t ϕt t 13

14 extends to a bounded oerator on L Λ τ 0 M ϕ will still denote the extension to L Λ τ 0 and we let ML Λ τ 0 stand for the set of all such mas. Then ML Λ τ 0 is a unital Banach algebra for the ointwise roduct and the following norm ϕ ML Λ τ 0 := M ϕ : L Λ τ 0 L Λ τ 0 Let M cb L Λ τ 0 be the subalgebra of all Fourier multiliers ϕ on L Λ τ 0 which are c.b. i.e. the corresonding oerators M ϕ are c.b., equied with the norm below ϕ := M ϕ : L Λ τ 0 L Λ τ 0 cb Mcb L Λ τ 0 By Pro. 0.4, a multilier ϕ belongs to M cb L Λ τ 0 if and only if the oerator M ϕ id S is bounded on L Λ τ 0 S as a subsace of L τ with ϕ = M ϕ id S Mcb L. Λ τ 0 By duality, it is very easy to see that for all 1, q where q ML τ 0 = ML q τ 0, M cb L τ 0 = M cb L q τ 0 = 1 we have isometrically. Note that the duality < f, g >= τ 0 fǧ for f in L τ 0, g in L q τ 0 and where ǧ := ĝt 1 t is the suitable choice to have the revious identifications via t G the identity ma. Therefore we can restrict ourselves to the case where. We see easily that ML τ 0 = M cb L τ 0 = l G isometrically. Since M cb L τ 0 ML τ 0 ML τ 0 = M cb L τ 0 contractively, we get by using the results of comlex interolation ML τ 0 ML τ 0 ML τ 0 M cb L τ 0 M cb L τ 0 ML τ 0 contractively. By reeating the same argument, we see that for all q < we have ML τ 0 ML q τ 0, M cb L τ 0 M cb L q τ 0 contractively. Thus are two decreasing families of algebras. ML τ 0 and M cb L τ 0 Now assume moreover that G is Abelian and equi its dual grou Ĝ which is comact with its Haar measure. In this case, the von Neumann algebra generated by G in B l G coincides with L Ĝ, L τ 0 coincides with L Ĝ and L τ coincides with L Ĝ, S. This alies e.g. for the grou Z which will be discussed later. Remark 0.10 It follows from well known results cf. e.g. [7], [8] that the canonical Hilbert transform defines a c.b. multilier on L for 1 < <. Therefore the natural roections of L onto L Λ which send f to fkz k are uniformly comletely bounded k Λ when Λ runs over all intervals of Z. In other words, the saces L Λ where Λ Z is an 14

15 arbitrary interval are uniformly comlemented in L as oerator saces. According to this, we see that for 1 < < the following inclusion mas M cb L Λ M cbl ϕ where ϕ is the trivial extension of ϕ equal to zero outside Λ are uniformly bounded when Λ runs over all intervals of Z. 0.9 Schur multiliers Let e kl k,l be the canonical basis of S, 1 and A be a subset of N N. A scalar ma ϕ defined on A is said to be a Schur multilier on S A if the associated oerator T ϕ : san e kl, k, l A san e kl, k, l A ϕ e kl ϕk, l e kl extends to a bounded oerator on S A T ϕ still denotes the extension of ϕ to S A and we let MS A stand for the set of all Schur multiliers on S A. Then MS A is a Banach algebra for the ointwise roduct and the norm ϕ := T MS ϕ : S A A S A We will denote by M cb S A the algebra of all Schur multiliers ϕ on S A which are c.b., equied with the norm ϕ = T ϕ : S Mcb S A A S A cb We will denote by M H S A and M cb HS A the subalgebras of MS A and M cbs A resectively formed of all Schur multiliers on S A which have a Hankelian form a multilier ϕ is viewed as an matrix. When A has a Hankelian form i.e. A = Λ for some set Λ N, we let MS A res. M cb S A be the algebra of all scalar mas ϕ defined on A such that the corresonding oerators ma S A boudedly res. comletely boundedly into itself. Note that a multilier on S A has necessarily a Hankelian form. For an examle of c.b. Hankelian Schur multiliers on S, we can quote the following. Fix z in T and consider the ma ϕ z : k, l z k+l. Then the corresonding oerator is by definition T ϕz : S S xkl k,l z k+l x kl where D z is the unitary oerator below D z = z z k,l = D zxd z

16 Clearly T ϕz is an isometry and in fact T ϕz is a comlete isometry. Therefore ϕ z belongs to Mcb HS for all 1. For the study of the saces MS and M cb S, we can again reduce to the case where since for 1, q such that = 1 we have q MS = MS q, M cb S = M cb S q isometrically. As noticed reviously, these identifications can be done via the identity ma if we wish by making a suitable choice for the duality between S and S q. Namely, we set x S, y S q < x, y >:= tr t xy There is a nice descrition of MS for and only for = and = 1 or =. Indeed MS = M cb S = l N N MS = M cb S = Γ l 1, l isometrically, where Γ l 1, l is the sace of all oerators from l 1 to l which factor through a Hilbert sace, equied with the usual factorization norm. The case = is trivial while the case = for which [3] gives the recise statement below and which goes in essence to Grothendieck is not see [3] for more references. Theorem 0.11 For ϕ : N N C, the following are equivalent. i ϕ is a Schur multilier on S with norm less than 1. ii There exist a Hilbert sace H, sequences of vectors h n n and k m m in the unit ball of H such that ϕn, m =< h n, k m > n, m in N. iii The oerator u ϕ : l 1 l which takes e n to m ϕn, me m belongs to Γ l 1, l with norm less than 1. Here e n n denotes the canonical basis of l 1. iv ϕ is a c.b. Schur multilier on S with c.b. norm less than 1. It is useful to note that this descrition rovides a class C of very simle multiliers in the unit ball of MS, those of the form ϕn, m = a n b m where a = a n, b = b n m m are in B l. The interest of this class C comes from the fact that there exists a universal constant K > 0 such that convc B MS KconvC where convc is the closure of the convex set generated by C in the simle convergence toology on N N cf. [3]. For < < we have contractive inclusions MS M cb S MS MS. Indeed MS embeds contractively into MS and we use the comlex interolation. More generally we see that for < < q < we have the following contractive embeddings MS MS q MS MS MS M cb S q M cb S MS Therefore MS and M cb S are two decreasing families of sets. 16

17 1 Non-commutative Λ-sets in discrete grous In this section G denotes an arbitrary discrete grou with unit e, denotes the left regular reresentation of G into B l G, L τ 0 denotes the non-commutative L -sace associated to the von Neumann algebra generated by G with resect to the usual trace τ 0 and L τ denotes the non-commutative L -sace associated to the von Neumann algebra generated by G Bl with resect to the trace τ = τ 0 tr where tr denotes the usual trace on the Schatten class S. Definition 1.1 Let < < be fixed and let Λ G be a given subset. We say that Λ is a Λ-set if the saces L Λ τ 0 and L Λ τ 0 are isomorhic, equivalently, there exists a constant λ > 0 such that for all finitely suorted families of scalars a t we have a t t λ L τ 0 a t We let λ Λ or sometimes simly λ stand for the smallest constant λ for which this haens. The reader is requested to see Subsection 0.8 for the definition of the algebra of Fourier multiliers ML τ 0 as well as its subalgebra M cb L τ 0. Definition 1. A set Λ G is said to be an interolation set for M L τ 0 for some 1 if the restriction ma below Q : ML τ 0 ϕ l Λ ϕt is µ-surective for some constant µ. We let µ Λ or simly µ be the smallest constant µ for which this haens. The following result shows that Λ-sets can be viewed as classes of interolation sets. Proosition 1.3 Let < < and Λ G be fixed. The assertions below are equivalent. i Λ is a Λ-set. ii Λ is an interolation set for M L τ 0. Moreover we have µ Λ λ Λ K µ L τ0 Λ where K L τ0 is the constant defined in the inequality 0.3. Proof: Assume that Λ is a Λ-set. For ε = ε t t in l Λ we let ε be its trivial extension to l G equal to zero outside Λ. Then for any f in L τ 0 L L ε t ft t = ε t ft t λ τ 0 τ 0 t G λ ε l Λ t G ft 1 = λ ε f l Λ 17 L τ 0 ε t ft ε λ f l Λ L τ 0

18 Thus ε is in ML τ 0 and it satisfies ε ML τ 0 λ ε l Λ. This means µ λ. Conversely assume that Λ is an interolation set for ML τ 0. Then for any δ > 0, each choice of signs ε on Λ admits a lifting ε in ML τ 0 with ε µ + δ. This ML τ 0 imlies that for any f in L Λ τ 0, say with finitely suorted Fourier transform f, we have L f µ + δ L τ 0 ε t ft t τ 0 since f = M ε M ε f. Then we integrate the inequality over all the choices of signs on Λ f L τ 0 µ + δ = µ + δ 1,1 Λ ε t ft t L τ 0 L ε t ft t 1,1 Λ,ν,L τ 0 dνε Now we aly the non-commutative version of Khintchine inequalities 0.3 and let δ tend to 0 to obtain f K µ L L τ0 f Hence Λ is a Λ-set with λ K τ 0 L τ 0. µ L τ0. Remarks 1.4 i Since the embeddings L τ 0 L q τ 0 L τ 0 L τ 0 are bounded for all real numbers < < q <, we see that the Λq-roerty imlies the Λ- roerty. Thus we have a decreasing family of sets Λ G Λ is a Λ-set. << ii Although no significantly new examles are known, it is useful to consider also the case 1 <. A set Λ is called a Λ-set in this case if L Λ τ 0 and L q Λ τ 0 are equivalent Banach saces for some and thus any 1 q <. Similarly we denote by λ Λ the smallest constant λ > 0 such that for any f in L Λ τ 0 we have f λf With this L τ 0 L 1 τ 0. terminology it is known using an extraolation argument that if q > and Λ is a Λq-set then Λ is a Λ-set. Conversely if Λ is a Λ-set then Λ is a Λq-set if and only if its indicator function 1l Λ belongs to ML q τ 0. Moreover for each set Λ we have ML λ Λ λ q Λ λ Λl Λ 1.5 q τ 0 Now we extend the revious definitions and results to the non-commutative case. Namely we define subsets of G laying for the sets M cb L τ 0 a rôle similar to the one layed by Λ-sets for ML τ 0. Definition 1.5 Let < < be fixed and let Λ G be a given set. We say that Λ is a Λ cb -set if there exists a constant C > 0 such that for all finitely suorted families of oerators x t in S we have L t x t C max τ x t x t 1 S, x t x t 1 S Then we let λ cb Λ stand for the smallest constant C for which the inequality above holds. 18

19 Remarks 1.6 i Using Jensen s inequality it is very easy to see that when, any f in L τ satisfies 1 max ft ft S, ft ft t G Hence the Λ cb -roerty means simly that the norms. on L Λ τ where for any f in L τ f := max t G t G S L τ f ft 1 ft S, ft ft t G L τ 1.6 and. are equivalent Therefore whenever Λ is a Λ cb -set the indicator function 1l Λ is in M cb L τ 0 i.e. the natural roection of L τ 0 onto L Λ τ 0 is c.b. with c.b. norm less or equal to λ cb Λ. ii Given two Λ cb -subsets Λ 1 and Λ of G, the set Λ 1 Λ has clearly the Λ cb -roerty with λ cb Λ 1 Λ λ cb Λ 1 + λ cb Λ. Definition 1.7 Given 1, a subset Λ of G is said to be an interolation set for M cb L τ 0 if the restriction ma below S Q : M cb L τ 0 ϕ l Λ ϕt is surective then it is µ-surective for some constant µ and we let µ cb Λ or simly µcb be the smallest constant µ for which this haens. The following result shows that in this more general setting, Λ cb -sets can be also viewed as classes of interolation sets. Proosition 1.8 Let < < and Λ G be fixed. The next assertions are equivalent. i Λ is a Λ cb -set. ii Λ is an interolation set for M cb L τ 0. Moreover we have µ cb Λ λcb Λ K L τ µcb Λ where K is the constant defined in L τ the inequality 0.3. Proof: Assume that Λ has the Λ cb -roerty. For ε = ε t t in l Λ we let ε be its extension to l G trivially by adding zeros. Then for any f = t G t x t in L τ, say with finitely many non-zero oerators x t, we have L ε t t x t λ cb max ε t x t x t τ S, ε t x t x t λ cb ε max l Λ x 1 tx t S, x t x t t G t G S 1 S 19

20 Hence using 1.6 we get L ε t t x t t G τ Thus ε is in M cb L τ 0 with ε Mcb L τ 0 λ cb ε λ cb ε l Λ l Λ f L τ which means that µ cb λcb. Conversely let Λ be an interolation set for M cb L τ 0 and δ be a fixed ositive number. Any choice of signs ε on Λ admits a lifting ε in M cb L τ 0 with ε µ cb + δ. Mcb L τ 0 This imlies since ε t = 1, t Λ that for any f = t x t in L Λ τ we have f L τ L µ cb + δ ε t t x t τ After letting δ tend to 0 we see that each f in L Λ τ satisfies for each choice of signs ε L f µ cb ε t t x t τ L τ We integrate the right inside of the inequality above over all the choices of signs on Λ f µ cb L ε t t x t dνε τ 1,1 Λ L τ L = µ cb ε t t x t 1,1 Λ,ν,L τ Now we aly the non-commutative version of Khintchine inequalities 0.3 to obtain 1 f K L L τ µ cb max x t x t τ S, x t x t S That is to say Λ is a Λ cb set with λ cb K L τ µcb. Remark. Since the embeddings M cb L τ 0 M cb L q τ 0 M cb L τ 0 ML τ 0 where < < q < are bounded, we see that the Λq cb -roerty imlies the Λ cb - roerty. Thus the family of sets Λ G Λ is a Λ cb -set is decreasing for << each fixed discrete grou G. On the other hand, the Λ cb -roerty trivially imlies the Λ-roerty. Moreover we have for any set Λ G and any < < q < λ cb Λ λ cb q Λ, µ Λ µ cb Λ, λ Λ λ cb Λ Comments 1.9 Clearly, we can naturally extend our definitions to the case of 1 <. We say that a set Λ G is a K cb -set if there exists a constant c > 0 such that for any sequence x t of oerators in S, say a finitely suorted sequence, we have c 1 inf y t yt 1 + zt z t t x tl S S τ 0

21 where the infimum runs over all decomositions of the x t s in S as x t = y t + z t. We let K cb Λ stand for the smallest constant c for which this holds. Recall that the converse inequality with constant 1 instead of c is satisfied by any set Λ and note that the K cb - roerty is trivial. Let 1 < < and = 1. Then, for a given set Λ G, the following are equivalent. i Λ is a K cb -set and each c.b. multilier on L Λ τ 0 extends to a c.b. multilier on L τ 0. ii Λ is a Λ cb -set. Indeed, assume i. By Pro. 1.8, we need to rove that Λ is an interolation set for M cb L τ 0 = M cb L τ 0. Equivalently, we need to rove that the choices of signs on Λ extend uniformly comletely boundedly to multiliers on L τ 0. Let ε = ε t be an arbitrary choice of signs on Λ. Then, for any finitely suorted sequence x t of oerators in S, we have L ε t t x t inf ε tx t=y t+z t y t y 1 t + zt z t τ y t,z t S S S = inf x t=y t+z t y t yt 1 + z t z t y t,z t S S S Thus by our assumtion on Λ we get L ε t t x t K cb Λ t x tl τ τ This means that ε defines a c.b. multilier on L Λ τ 0 with ε Mcb L Λ τ 0 Kcb Λ. The second assumtion on Λ says that the restriction ma M cb L τ 0 M cb L Λ τ 0 ϕ ϕ Λ is surective, hence µ-surective for some constant µ > 0. Then, for all δ > 0, each choice of signs ε on Λ extends to a c.b. multilier on L τ 0 with norm less or equal to µk cb Λ+δ. Therefore we are done and we have µ cb Λ = µcb Λ µkcb Λ. Conversely, assume ii. Then by Pro. 1.8, Λ is an interolation set for M cb L τ 0 = M cb L τ 0. A fortiori, each c.b. multilier on L Λ τ 0 extends to a c.b. multilier on L τ 0 since every multilier on L Λ τ 0 is in articular a bounded sequence on Λ. On the other hand, let δ > 0 be fixed. Then since every choice of signs ε on Λ admits a lifting ε with ε µ cb Λ + δ, we get for every f = t x t in L Λ τ Mcb L τ 0 L ε t t x t µ cb L Λ + δ t x t τ τ Hence if we let δ tend to zero, integrate over all these choices of signs and aly the non-commutative version of Khinchine s inequalities 0., we obtain 1 K L τ y t yt + zt z t µ cb Λ L t x t S τ inf x t=y t+z t y t,z t S 1 S

22 where K L τ is the constant of 0.. This means that Λ is a K cb -set. Moreover we have K cb Λ µcb Λ = µcb Λ. K L τ K L τ Remark. Let 1 1 < < < <. Then, it is well known that any Λ -set is a Λ 1 -set. However, we do not know whether a Λ cb -set must necessarily be a K 1 cb - set. In the sequel, we are interested in finding roerties simler and stronger than the Λ cb - roerty. This aim is achieved for even integers by introducing two combinatorial roerties called the B and the Z-roerties. For the grou Z, the B-roerty was firstly considered by W. Rudin in [37] while the Z-roerty was introduced by A. Zygmund in his work [46] which ustifies the name of the latter roerty. Thus our roerties are nothing but an adatation of Rudin s roerty to the case of arbitrary discrete grous, and a generalization of Zygmund s roerty for arbitrary ositive integers and arbitrary discrete grous. Definition 1.10 Let be an arbitrary integer. A subset Λ G has the B-roerty if for all -tules t 1, t,..., t and s 1, s,..., s in Λ, t 1 1 s 1t 1 s...t 1 s = e e is the unit of G holds if and only if t 1, t,..., t = s 1, s,..., s 1. Examle. When G is the free grou, every free subset Λ of G has the B-roerty. Indeed, let t 1, t,..., t, s 1, s,..., s in Λ be such that t 1 1 s 1 t 1 s...t 1 s is the emty word and assume t 1, t,..., t s 1, s,..., s. Denote by i 0 the first index such that t i0 s for all 1 and i 1 the last index for which t i0 = t i1. Then, we would have t 1 i 0 s i0...s i1 1t 1 i 1 = s 1 i t 1s 1...t i1 +1s 1 i 1. The reduced word of t 1 i 0 s i0...s i1 1t 1 i 1 is exressed with letters in Λ and contains necessarily the letter t 1 i 0 while the reduced word of s 1 i t 1s 1...t i 1 +1s 1 i 1 which is also exressed with letters in Λ does not contain the letter t 1 i 0. This means that there exists a word which has two different reduced exressions, both with letters belonging to Λ which contradicts the freeness of Λ. We will see later that this roerty which is adated to free grous is well adated to the case of the grou Z. Definition 1.11 Let be an arbitrary integer. For each 1 i, we set ν i = 1 when i is even and ν i = 1 otherwise. Then, we say that a set Λ has the Z-roerty if Z Λ <, where Z Λ := su t 1, t,..., t Λ i, ti t & t ν 1 1 t ν...t ν = γ γ G Proosition 1.1 Let < be an arbitrary integer. Then, the B-roerty imlies the Z-roerty. Moreover, each B-subset Λ of a discrete grou G satisfies Z Λ! if is even and Z Λ +1! if is odd. 1 In each set, elements are reeated a number of times equal to their multilicity in the corresonding sequence.

23 Proof: Let t 1, t,..., t, s 1, s,..., s be -tules in Λ such that t ν 1 1 tν...tν = s ν 1 1 sν...sν with ν 1 as in Definition 1.11 and t i t, s i s for all 1 i. Then, we get t ν 1 1 tν...tν s ν...s ν s ν 1 1 = e. Since Λ has the B-roerty, we have necessarily t i ; 1 i, i odd s i ; 1 i, i even = t i ; 1 i, i even s i ; 1 i, i odd But t i t and s i s for all 1 i, therefore we have t i ; 1 i, i even = s i ; 1 i, i even t i ; 1 i, i odd = s i ; 1 i, i odd and it is easy to deduce from this the announced control of the constant Z Λ. Theorem 1.13 Let < be an integer and let G be a discrete grou. Then, every subset Λ of G with the Z-roerty is a Λ cb -set. Moreover, there exists a constant C deending only on such that for each set Λ G, we have λ cb Λ 3 maxz Λ, C. The roof of Theorem 1.13 is much easier to follow in the articular case = for which the roof aears in the Aendix Pro. 6.1 and we urge the reader to look at it first before studying the roof of Theorem For the roof of Theorem 1.13, it will be convenient to set the following definitions and to use the inequality of Pro which was observed by G. Pisier. Given a artition P of 1,,...,, we set k l P for each 1 k, l if k and l belong to a same element in P. Now given two artitions P 1 and P of 1,,...,, we set P 1 P if for each 1 k, l, k l P 1 whenever k l P and we set P 1 < P if P 1 P and P 1 < P. This rovides the set of all the artitions on 1,,..., with a artial order for which P max = 1,,..., is a unique maximal element and P min = 1,,..., is a unique minimal one. Finally, the artition P ξ on 1,,..., associated to a given -tule ξ = ξ 1, ξ,...ξ in I where I is an arbitrary set, is defined as the unique artition such that for all 1 k, l, k l P ξ if and only if ξ k = ξ l. Proosition 1.14 Let Ω, Σ, P be a robability sace and ε i i I a family of indeendent random variables with Pε i = 1 = 1 = Pε i = 1 for each i I. Let be an arbitrary integer and let for 1, E be Banach saces, f : I E be finitely suorted functions and ϕ : E 1 E... E F be a -linear ma of norm less or equal to 1, where F is a given Banach sace. Fix a artition P of the set 1,,..., and set A P := 1,,..., P. Then we have ξ I,P ξ P ξ=ξ 1,ξ,...,ξ ϕ f 1 ξ 1, f ξ,..., f ξ F A P i I f i E A P 1 Ω i I ε i f i dp E An element is reeated in a given set a number of times equal to its multilicity in this set. 3

24 Proof of Pro. 1.14: We start by noticing the following. Given a finite set of indexes α = 1,,..., s with s elements s, we consider s 1-indeendent coies of the family ε i i I on Ω, Σ, P assumed large enough, denoted by Y 1 α, i, Y i I α, i, i I..., Y s 1 α, i. Then we set i I Z 1 α, i = Y 1 α, i Z k α, i = Y k 1 α, iy k α, i, k s 1 Z s α, i = Y s 1 α, i Clearly, each of the families Z 1 α, i, Z i I α, i,..., Z i I s α, i has the same i I distribution as the family ε i. Moreover, using successively the orthonormality of each i I of the families Y k α, i, we check easily that for any function η : α I, the integral i I s Z k α, ηk dp is equal to 1 if the function η is constant on α and 0 otherwise. Ω k=1 Now if we are given a artition P of 1,,..., say P = α 1, α,...α N then for each set α k 1 k N with α k, we can define as above a family Z α k, i. Moreover we i I α k can construct these families so that they are mutually indeendent. A simle verification shows that ξ I,P ξ P ξ=ξ 1,ξ,...,ξ ϕ f 1 ξ 1, f ξ,..., f ξ = Ω ϕ φ 1 ω, φ ω,..., φ ω dpω where we have set for each integer 1 Z α k, iωf i if α k with α k i I ω Ω, φ ω := f i if A P. Hence by using Hölder s inequality we get ϕ f 1 ξ 1, f ξ,..., f ξ F ξ I,P ξ P ξ=ξ 1,ξ,...,ξ φe A P A P 1 Ω i I φ E dp = f i E = A P f i i I A P E i I A P 1 Ω φ1 ω E1 φ ω E... φ ω E dpω Ω A P 1 Ω Z α k, if i dp E i I ε i f i dp E Proof of Theorem 1.13: Let f = t x t with t x t finitely suorted. Then f L τ = τ f f = f µ 1 f µ...f µ L τ i I = γ 4 γ G ξ 1,ξ,...,ξ Λ ξ ν 1 1 ξν...ξν =γ x µ 1 ξ 1 x µ ξ...x µ ξ 1 L τ

25 where, for each 1 k, we have set µk = 1, ν k = 1 if k is even µ k =, ν k = 1 if k is odd. Then we have f L τ γ G = γ G ξ Λ,P ξ =P max ξ=ξ 1,ξ,...,ξ ξ ν 1 1 ξν...ξν =γ ξ 1,ξ,...,ξ Λ ξ ν 1 1 ξν...ξν =γ x µ 1 ξ 1 x µ ξ...x µ ξ x µ 1 ξ 1 x µ ξ...x µ ξ + C S = S γ G P artition of1,,..., P P max P artition ξ Λ,P ξ =P of 1,,..., ξ=ξ 1,ξ,...,ξ ξ ν 1 1 ξν...ξν =γ γ G ξ Λ,P ξ =P ξ=ξ 1,ξ,...,ξ ξ ν 1 1 ξν...ξν =γ x µ 1 ξ 1 x µ ξ...x µ ξ x µ 1 ξ 1 x µ ξ...x µ ξ where C is a constant deending only on and more recisely on the number of artitions of the set 1,,...,. Henceforth, all the constants which will aear during the roof and which deend on only will be denoted by C for simlicity. On the other hand, let S := max x t x t and for each artition P, let SP := γ G S, ξ Λ,P ξ =P ξ=ξ 1,ξ,...,ξ ξ ν 1 1 ξν...ξν =γ With these notations, the inequality above becomes f SP max + C L τ x t x t S x µ 1 ξ 1 x µ ξ...x µ ξ S SP.7 P artition of 1,,...,,P P max Our aim is to rove that SP max Z ΛS and that there exists C such that for each artition P P max, we have SP C S f 1.8 L τ Ste 1. The assumtion Λ has the Z-roerty ensures SP max Z ΛS. Indeed SP max = γ G ξ 1,ξ,...,ξ Λ ξ i ξ, 1 i ξ ν 1 1 ξν...ξν =γ Z Λ = Z Λ x µ 1 ξ 1 x µ ξ...x µ ξ ξ 1,ξ,...,ξ Λ ξ 1,ξ,...,ξ Λ S Z Λ γ G x µ ξ...x µ ξ x µ 1 x µ ξ 1 x µ 1 ξ...x µ ξ x µ 1 ξ 1 x µ 1 5 ξ 1,ξ,...,ξ Λ ξ i ξ, 1 i ξ ν 1 1 ξν...ξν =γ ξ 1 x µ ξ 1 x µ ξ...x µ S ξ 1 ξ...x µ S ξ 1 S x µ 1 ξ 1 x µ ξ...x µ ξ S S

26 Z Λ i=1 max ξ i Λ x S ξ i x ξi, x ξi x S ξ i where for the last inequality we alied Corollary 0.9. Therefore SP max Z ΛS. Ste. Given an integer 1 k, we show that if 1.8 is satisfied for all the artitions P with P k, then it is also satisfied for all the artitions P with P k+1. Indeed, let P 0 be a fixed artition with P 0 = k + 1, then SP 0 = γ G SP 0 γ G γ G ξ Λ,P ξ P 0 ξ=ξ 1,ξ,...,ξ ξ ν 1 1 ξν...ξν =γ ξ Λ,P ξ P 0 ξ=ξ 1,ξ,...,ξ ξ ν 1 1 ξν...ξν =γ ξ Λ,P ξ P 0 ξ=ξ 1,ξ,...,ξ ξ ν 1 1 ξν...ξν =γ x µ 1 ξ 1 x µ ξ...x µ ξ x µ 1 ξ 1 x µ ξ...x µ ξ x µ 1 ξ 1 x µ ξ...x µ ξ ξ i Λ ξ Λ,P ξ <P 0 ξ=ξ 1,ξ,...,ξ ξ ν 1 1 ξν...ξν =γ + S γ G S + C x µ 1 ξ 1 x µ ξ...x µ ξ ξ Λ,P ξ <P 0 ξ=ξ 1,ξ,...,ξ ξ ν 1 1 ξν...ξν =γ P artition of 1,,..., P<P 0 S x µ 1 ξ 1 x µ ξ...x µ ξ SP According to the induction hyothesis, each P < P 0 satisfies 1.8 since its cardinal satisfies P < P 0 = k + 1, hence we are reduced to rove the following inequality x µ 1 ξ 1 x µ ξ...x µ ξ C S f S L τ γ G ξ Λ,P ξ P 0 ξ=ξ 1,ξ,...,ξ ξ ν 1 1 ξν...ξν =γ = γ G ξ Λ,P ξ P 0 ξ=ξ 1,ξ,...,ξ ξ Λ,P ξ P 0 ξ=ξ 1,ξ,...,ξ ξ ν 1 1 ξν...ξν =γ x µ 1 ξ 1 x µ ξ...x µ ξ S = γ γ G ξ Λ,P ξ P 0 ξ=ξ 1,ξ,...,ξ ξ ν 1 1 ξν...ξν =γ x µ 1 ξ 1 x µ ξ...x µ ξ ξ1 x ξ1 µ1 ξ x ξ µ... ξ x ξ µ = ξ Λ,P ξ P 0 ξ=ξ 1,ξ,...,ξ f 1 ξ 1 f ξ...f ξ L τ where for each 1, f is defined on Λ by setting f t = t x t µ, t Λ. The f s belong to L τ. At this level, we aly Pro to 1, 1 N equied with the counting robability ν, to the n t -th coordinate roection on 1, 1 N denoted by ε t where n t, t Λ is an enumeration of the set Λ, to the functions f : Λ L τ above and to the -linear contractive ma which is the roduct from L τ L τ... L τ times into L τ. Hence, letting A P0 := 1,,..., P 0, we get ξ Λ,P ξ P 0 ξ=ξ 1,ξ,...,ξ f 1 ξ 1 f ξ...f ξ 6 L τ L τ S L τ