Type, cotype and K-convexity

Bernard Maurey

Laboratoire d’Analyse et Mathématiques Appliquées, UMR 8050

Université de Marne la Vallée

Boulevard Descartes, Cité Descartes, Champs sur Marne 77454 Marne la Vallée Cedex 2, FRANCE

Preprint submitted to Elsevier Preprint 14 October 2002

1 The pre-history of type and cotype, as I remember it

At the end of the sixties, Pietsch [Pi] promoted the notion of p-summing operators between Banach spaces, which extends to all values of p ∈ [1, +∞) the study of some classes of operators introduced by Grothendieck [Gro], under different names, for the special values p = 1, 2. In an important paper devoted to p-summing operators, Lindenstrauss and Pełczyński [LP] gave a second birth to what we know in Banach space theory as the Grothendieck theorem; one formulation of it states that every operator from ℓ_{1} to ℓ_{2} is 1-summing; another formulation is the famous Grothendieck’s inequality. Around 1969, L. Schwartz introduced radonifying maps, a notion that turned out to be closely related to p-summing maps. A special case of this notion deals with the Wiener measure and with linear maps from a Hilbert space H to a Banach space X, that transform the canonical cylindrical Gaussian measure of H into a true Radon probability measure on X (see L. Gross [Gr1,Gr2] for another viewpoint on this subject). L. Schwartz organized a seminar at the Ecole Polytechnique in Paris ([Sem], 1969–70) about these topics. This is one of the reasons why Paris, and especially the Ecole Polytechnique, became one of the places where the subject of type and cotype was developed.

Type and cotype conditions appeared first in the framework of p-summing operators, or more precisely in connection with the factorization through L_{p}, p > 1, of operators with values in L_{1} (in this paper, operator means bounded linear operator). In the spring of 1972 I saw the preprint of the paper [Ro] by H. Rosenthal; this paper played an essential role for me; it contains several ideas that I later used and developed in [Ma2]. Two of these ideas taken from [Ro] are the factorization conditions and the notion of stable type p. By Pietsch’s factorization theorem, which extends some factorization results due to Grothendieck [Gro], every q-summing operator from C(K) to a Banach space factors through the natural injection C(K) → L_{q}(K, μ), for some probability measure μ on K. Rosenthal dualizes this fact, and shows that given T : X → L_{1} linear such that T^{∗} is q-summing, then T factors through a multiplication operator M_{f} : L_{p} → L_{1} by a function f ∈ L_{q} (1/p + 1/q = 1; let us simply write L_{r} for L_{r}(K, μ), 0 < r ≤ +∞); we have thus T = M_{f} ∘ T_{1}, where T_{1} : X → L_{p} is bounded and linear. One can give direct conditions on T that guarantee this factorization, with no need to further reference to q-summing maps: if an operator T : X → L_{1} is such that

∫ (∑_{i} |T (x_{i})|^{p})^{1/p} dμ ≤ C (∑_{i} ∥x_{i}∥^{p})^{1/p},

for some C and every finite sequence (x_{i}) ⊂ X, then T factors as T = M_{f} ∘ T_{1} for some f ∈ L_{q}. The proof of the factorization theorem is just an application of the Hahn-Banach separation theorem, either directly as in [Ma1], or by going back to Pietsch’s factorization as in [Ro]. One gets in this way a function f ∈ L_{q} such that ∥f∥_{q} ≤ 1 and ∫ |T (x)/f|^{p} dμ ≤ C^{p} ∥x∥^{p} for every x ∈ X. The above operator T_{1} is then defined by T_{1}(x) = T (x)/f ∈ L_{p} for every x ∈ X. Next, it is shown in [Ro] that a simple norm condition on X, that happens to be true for X = L_{s} when 2 ≥ s > p > 1, easily implies the above factorization condition, as soon as T : X → L_{1} is bounded (and linear). This condition on a Banach space X is of the form

∫ ∥∑_{i} f_{i}(t)x_{i}∥ dt ≤ K (∑_{i} ∥x_{i}∥^{p})^{1/p},

where K is a constant depending only upon X, (f_{i}) is a sequence of L_{1}- normalized p-stable variables, and (x_{i}) an arbitrary sequence in X. This condition was called stable type p in [Ma1,Ma2]; it was used in [Ro] (without this name) for the injection of X ⊂ L_{1} to L_{1}, and in the general case in [Ma2]. For example, since a Hilbert space has type 2, we obtain in this way that every bounded linear map from a Hilbert space to L_{1} factors through a multiplication M_{f} : L_{2} → L_{1}, a statement dual to one of the results of [Gro]: every operator from a C(K)-space to a Hilbert space is 2-summing. By trace duality, this yields that every operator from ℓ_{1} to ℓ_{2} is 2-summing; we may call this the easy Grothendieck theorem. The same proof shows that every operator from a C(K)-space, to a space X such that the dual X^{∗} has type 2, is 2-summing: this result appeared for the first time in a paper by Dubinsky, Pełczyński and Rosenthal [DPR].

It is obvious to generalize to operators from X to L_{r} the condition that gives a factorization through a multiplication operator L_{p} → L_{r} (0 < r < p, see [Ma1,Ma2]). In particular, some of the results obtained for 0 < r < 1 are parallel to results obtained earlier by Nikišin [N1,N2]: since every Banach space X has stable type 1 − ε for every ε > 0, every operator from X to L_{r}, 0 < r < 1, factors through L_{1−ε} when 1 − ε ≥ r.

A first relation between these topics and finite dimensional geometry comes from the paper [Ro]; there, a delicate quantitative Lemma (Lemma 6 from [Ro]) shows that when the injection from a subspace X ⊂ L_{1} to L_{1} does not factor through any L_{p}, p > 1, then X must contain complemented almost isometric copies of ℓ_{1}^{n} for every n ≥ 1, proving thus that every reflexive subspace of L_{1} embeds in some L_{p}, p > 1 (the main result of [Ro]). This Lemma was extended in [Ma2] to a general Banach space X as follows: when there exists an operator T : X → L_{p} that does not factor through any L_{p+ε}, ε > 0, then the injections ℓ_{1}^{n} → ℓ_{p}^{n}, n ∈ ℕ, uniformly factor through X. In particular, when there exists an operator T : X → L_{1} that does not factor through any L_{1+ε}, ε > 0, then X contains uniformly isomorphic and complemented copies of ℓ_{1}^{n}, for every n ≥ 1. This gives a new (bizarre) proof of Grothendieck’s theorem: since ℓ_{1}^{n} is not uniformly complemented in c_{0}, the preceding statement implies that every bounded linear map from c_{0} to L_{1} factors through L_{1+ε}, and it reduces Grothendieck’s theorem to a much easier variant. It is a model for a list of reduction results, for example this sort of extension of the Grothendieck theorem: every operator from a cotype 2 space X to any Banach space, which is 2-summing, is already 1-summing (see [Ma2]; as we have just said, when X = L_{1}, this is the information that one needs in order to pass from the easy Grothendieck theorem to the real one). This line of results displayed interesting connections between some simple finite dimensional phenomenons and analytic facts about Banach spaces.

In the same years, Hoffman-Jørgensen [HJ1] proved general results about series of vector valued independent random variables, that are in the spirit of Kahane’s inequalities for vector valued Rademacher series; he also defined Rademacher type-p and showed connections to the law of large numbers in [HJ2]. The notion of type 2 (with a different name) appeared first in [DPR], and it was shown in this article that stable type 2 and Rademacher type 2 are identical. The results from [HJ1] imply that stable type p and Rademacher type p are closely related for every p ∈ (1, 2]: stable type p implies Rademacher type p, and Rademacher type p implies stable type p−ε for every ε > 0. Later on, it has been universally admitted that Rademacher type is easier to work with, and the notion of stable type p essentially disappeared, except for p = 2, because 2-stable type and cotype express interesting properties of Gaussian probability measures on a Banach space. With Rademacher type p (we say simply type p in what follows), several points are simplified; it is obvious that type p implies type r for r ≤ p, and the opposite for cotype; the results for L_{r} spaces are easier to formulate, and simple to prove using Khintchine’s inequality: L_{r} has type r and cotype 2 when 1 ≤ r ≤ 2 and type 2 and cotype r when 2 ≤ r < +∞. Clearly, L_{r} does not have type r + ε, ε > 0 when 1 ≤ r ≤ 2, and does not have cotype r − ε when 2 ≤ r ≤ +∞. This suggested that one could possibly read some geometrical information about X from the limit values of p and q that give type p or cotype q for X.

The first attempts to relate type, cotype to the fact that X contains almost isometric copies of some classical spaces concerned ℓ_{∞}^{n} and ℓ_{1}^{n}. The first result [MP1] gave the equivalence between non-trivial cotype for X and the fact that X does not contain ℓ_{∞}^{n} uniformly; today, the proof in [MP1] looks a bit ridiculous by its complication. It was presented at the Conference at Oberwolfach, October 73; at the same meeting, James presented a much deeper result, namely his solution of the “reflexive vs B-convex” problem (see below). This was perhaps the beginning of what was later called “Local theory”. For the relation between the absence of ℓ_{1}^{n}s in a Banach space X and other properties of this X, the first steps are due to Beck, Giesy and James, several years before this story [Be,G1,J1]; Beck showed the relevance to the law of large numbers in Banach spaces of the fact that X does not contain copies of ℓ_{1}^{n}s. Beck and Giesy defined B-convex Banach spaces as follows: the Banach space X is Bconvex if for some n > 1 and ε > 0, and for all norm one vectors (x_{i})_{i=1}^{n} in X, at least one choice of signs gives ∥ ∑_{i=1}^{n} ±x_{i}∥ ≤ n (1 − ε). Giesy proved several Banach space flavoured results about B-convexity, for example that X^{∗} and X^{∗∗} are B-convex when X is B-convex. James [J1] also worked on this class, which he called uniformly non ℓ_{1}^{n}; in this paper [J1], he conjectured that Bconvex spaces can be renormed to be uniformly convex, and must therefore be reflexive (and he disproved this conjecture in 1973, as we have said above).

Shortly after the result for cotype and ℓ_{∞}^{n}, Pisier proved the type and ℓ_{1}^{n} case [P1]; he developed the submultiplicativity method for the type constants, which was important for the following paper [MP2]. Pisier’s result showed that the class of B-convex spaces coincides with the class of spaces X that have type p for some p > 1. Then Pisier and I started to work on the relations between the limit values for the type or cotype of X, and the existence of subspaces of X that look somewhat like ℓ_{p}^{n}. Our first approach to the results of [MP2] was to strengthen the Dvoretzky-Rogers factorization [DR] for a Banach space X, using information on the limits of type and cotype; it just happened that the beautiful result of Krivine [Kr2] (see section 4) appeared during the preparation of [MP2] and allowed us to prove a much more satisfactory result. In the first version of [MP2], we proved that when X has type p − ε but not p + ε for every ε > 0, then the injections ℓ_{1}^{n} → ℓ_{p}^{n} factor almost isometrically through a subspace of X for all n ≥ 1, which means that we can find norm one vectors x_{1}, …, x_{n} in X such that

(∑_{i=1}^{n} |a_{i}|^{p})^{1/p} ≤ (1 + ε) ∥∑_{i=1}^{n} a_{i}x_{i}∥ ≤ (1 + ε) ∑_{i=1}^{n} |a_{i}|

for all scalars (a_{i}); the second inequality is of course obvious. When p < 2, this is a strengthening of the Dvoretzky-Rogers Lemma which says that the above statement holds in every Banach space when p = 2. Krivine’s theorem appeared shortly after the first version of [MP2] was written; fortunately, Studia Math was so slow to publish at that time that we were able to modify our article in the form which is known as Maurey-Pisier or Maurey-Pisier-Krivine theorem. I will call it here MP+K theorem, to emphasize the fact that these three persons did not work together on this particular paper.

Kwapień was visiting Paris in 1971 and 72, just before all this started, and he played a significant role in the mathematical education of some of the young French; he gave several seminar talks that had a serious impact on us; he read and found the mistakes in several false “new proofs” that I had for the Grothendieck theorem, and he was the first person who checked the eventually correct proof of that I gave in [Ma2]. His result in [Kw] had a great influence on the subject of type and cotype; it appeared actually before the definitions of type and cotype were given, but it is nice to formulate it as follows: if X has both type 2 and cotype 2, then X is isomorphic to a Hilbert space. This is one of the first isomorphic characterizations of the Hilbert space. Some time later, I used in [Ma3] a small modification of Kwapień’s argument and showed that every bounded linear operator from a subspace X_{0} of a type 2 space X to a cotype 2 space Y factors through a Hilbert space, and extends to an operator from the whole space X to Y. In particular, every cotype 2 subspace X_{0} of a type 2 space X is Hilbertian and complemented in X. This was a generalization of a well known result due to Kadec and Pełczyński [KP], that Hilbertian subspaces of L_{p}, 2 ≤ p < +∞, are complemented.

Super-properties appeared in the work of James on super-reflexivity (see [J2] and [J3], and section 2 below); ultraproduct methods [DK] give more insight on super-properties: a property is a super-property when it passes to ultrapowers. Super-reflexivity is obviously a super-property, and B-convexity is another super-property; James showed that super-reflexive spaces are B-convex. Deciding whether B-convex and super-reflexive spaces are the same class, as was conjectured by James in [J1], remained a difficult problem for some time, and was finally solved by James, who constructed a non-reflexive B-convex space ([J4], improved in [J5]); before this, Brunel and Sucheston [BS1,BS2] had tried to prove that B-convex spaces were reflexive, and a part of their attempt introduced an important concept, that of spreading model, which will be used here in sections 4 and 5. From this point on, there were two clearly distinct settings: super-reflexive spaces are those that can be renormed to be uniformly convex (Enflo [En]); they have martingale type p (the basis for Pisier’s renorming theorem [P2]), and the class of B-convex or type-p spaces, p > 1, is strictly larger. However, contrary to the general case, type and uniform convexity are strongly related for lattices (see Johnson [Jo], and [LT, 1.f]). In a lattice X with non-trivial cotype, it is possible to prove Khintchinetype inequalities. Given (x_{i})_{i=1}^{n} in X, these inequalities permit to replace the estimate of a Rademacher average ∑_{i=1}^{n} ε_{i}(t)x_{i} in L_{2}(X) by an estimate of the square function (∑_{i=1}^{n} |x_{i}|^{2})^{1/2} in X. This kind of “functional calculus” for lattices was developed by Krivine in [Kr1], where he obtained interesting formulations of the Grothendieck theorem, relating operators between lattices and the square function (see also [LT, 1.f.14]).

Early signs of a tendency to move from abstract Banach spaces to the study of C^{∗}-algebras and operator spaces also came in this framework. N. Tomczak [To] proved that the Schatten classes have the same type or cotype properties than the L_{p} spaces. Pisier [P3] generalized Grothendieck’s theorem to C^{∗}- algebras; the result was revisited by Haagerup [Ha] and was the start for many further exchanges between them. Several other factorization results related to Grothendieck’s theorem were proved in those years, see [P8].

The first really striking application of cotype as a classification tool appears in the results of Figiel, Lindenstrauss and Milman [FLM]. They showed that Dvoretzky’s theorem takes a very strong form in cotype 2 spaces: if X has cotype 2, there exists a constant c > 0 such that for every integer n, every ndimensional subspace of X contains a further subspace X_{0} such that dim X_{0} = m ≥ c n and d(X_{0}, ℓ_{2}^{m}) ≤ 2. This result makes use of a certain fundamental formula

k = [η(τ) n M_{r}^{2}/b^{2}]

proved in [FLM, Theorem 2.6], relating the dimension k of (1 + τ)-spherical sections of an n-dimensional normed space to some integral invariant M_{r}. This formula appears already –with a different normalization– as equation (14) in [Mi1]. It gives the spectacular consequence above when using cotype 2 in an appropriate way; actually, [FLM] quantify the dimension of spherical sections in terms of the cotype q property, for every q ≥ 2, and the previous result for cotype 2 is a special case. Another approach to the problem of spherical sections, the notion of volume ratio developed by Szarek and Tomczak [ST], also singles out the special behaviour of cotype 2 spaces. This approach is based on the work of Szarek [Sz], who introduced volume arguments in a new proof of the results of Kašin [Kš] about ℓ_{1}^{n}; of course Szarek need not mention cotype 2 when working with the explicit norm of ℓ_{1}^{n}! The fact that cotype 2 spaces have a uniformly bounded volume ratio was proved later by Bourgain and Milman [BMi], and this motivated the introduction of weak cotype 2 by Milman and Pisier ([MP], see also Chapter 10 of Pisier’s book [P9]).

Type is a nice tool for estimating the behaviour of the entropy of a convex hull; a simple observation of mine, written in [P5], was used in entropy problems by Carl [Ca]. This observation states that in a Banach space X with type p > 1, every point x from the convex hull of a subset A of the unit ball B_{X} can be approximated by a convex combination of n points of A, with an error of order n^{−1/q} (with q conjugate to p). Lemma 9 below is in the spirit of this result.

Type and cotype have some simple stability properties; for example, the dual of a type p space has cotype q for the conjugate exponent, but the converse is false as shown by the pair (ℓ_{1}, ℓ_{∞}). The two young and ignorant authors of [MP2] left open a nice intriguing conjecture: is it possible to dualize cotype when we have some non-trivial type? It is clear that what is needed is the boundedness of the Rademacher projection on L_{2}(X). Spaces such that the Rademacher projection is bounded were called K-convex in [MP2] (was it because K was the first available letter after J for J-convex, a notion due to James and named by Brunel and Sucheston [BS2], or to acknowledge the importance of Kwapień’s work on Rademacher averages?) It was conjectured in [MP2] that every space with type r > 1 should be K-convex, which would imply that the dual X^{∗} of a space X with cotype q and some non-trivial type should be of type p, with 1/p + 1/q = 1. Six years later, Pisier proved what I consider the most beautiful result in this area, making use of Kato’s theorem on holomorphic semi-groups (see [P6] and section 6 of this article): every B-convex space is K-convex.

Although very beautiful, the preceding theorem is not the one that has been most useful for local theory. The most useful is another result obtained earlier by Pisier [P4], on the way to the general theorem above. This result asserts that the K-convexity constant of X is bounded by C (1 + ln d_{X}), where d_{X} is the distance from X to the Hilbert space of the appropriate dimension (see Theorem 13 below). In particular, the K-convexity constant is bounded by C (1 + ln n) for any n-dimensional normed space. The quantitative finitedimensional K-convexity, together with the notion of ℓ-norm, leads to a powerful tool for geometric estimates (Theorem 3.11 in [P9]; this theorem appeared first in [FT]). These results play an important role in the QS-theorem of Milman ([Mi2], see also [P9]).

2 Super-properties

Several of the properties P that are defined for a Banach space X are expressed in the following way: suppose that a number N_{P} (E) is associated to every finite dimensional normed space E, in such a way that N_{P} (F) tends to N_{P} (E) when the Banach-Mazur distance d(F, E) between F and E tends to 1; the most common such dependence is when N_{P} (F) ≤ d(F, E) N_{P} (E). We then say that the Banach space X satisfies property P when N_{P} (X) = sup_{E} N_{P} (E) < +∞, where the supremum is extended to all finite dimensional subspaces E of X.

Clearly, the fact that such a property P holds for X only depends upon the family F(X) of all finite dimensional normed spaces F such that for every ε > 0, there exists E ⊂ X for which d(F, E) < 1 + ε. After James [J2], we say that Y is finitely representable in X when F(Y) ⊂ F(X); for instance, L_{p} is finitely representable in ℓ_{p}, and it is known that X^{∗∗} is finitely representable in X for every Banach space X (local reflexivity). A property P of Banach spaces is called super-property if we know that whenever a Banach space X has P, then every Banach space Y finitely representable in X has P. Clearly, every property P expressed by N_{P} (X) < +∞ as above is a super-property. Super-properties were defined by James in [J2].

Type and cotype are such properties. Let us recall a few definitions and facts that are developed in [JL]. Let (ε_{i})_{i=1}^{+∞} denote the sequence of Rademacher functions on [0, 1], or any independent sequence of centered Bernoulli random variables. Let p ∈ [1, +∞). We say that X has type p when there exists a constant T such that

(∫_{0}^{1}∥∑_{i=1}^{n} ε_{i}(t) x_{i}∥^{2} dt)^{1/2} ≤ T (∑_{i=1}^{n} ∥x_{i}∥^{p})^{1/p},

for every n ≥ 1 and every sequence (x_{i})_{i=1}^{n} ⊂ X; we denote by T_{p}(X) the smallest constant T with this property; obviously, every normed space X has type 1 with T_{1}(X) = 1. On the other hand, it follows from Khintchine’s inequalities that no non-zero normed space has type p when p > 2. Saying that X has type p is obviously equivalent to the fact that the family of finite dimensional subspaces E of X satisfies sup_{E} T_{p}(E) < +∞, thus having type p is a super-property. We say that X has cotype q when there exists a constant C_{q}(X) such that

(∑_{i=1}^{n} ∥x_{i}∥^{q})^{1/q} ≤ C_{q}(X) (∫_{0}^{1}∥∑_{i=1}^{n} ε_{i}(t) x_{i}∥^{2} dt)^{1/2}

for every n ≥ 1 and every sequence (x_{i})_{i=1}^{n} ⊂ X; again, this is equivalent to the fact that sup_{E} C_{q}(E) < +∞, and cotype is therefore another super-property. In both definitions of type and cotype, the choice of the L_{2} norm for the Rademacher averages is irrelevant (except for the exact value of the constants); this follows from Kahane’s inequalities (see [Ka, Chapter II, Th. 4]), which state that for every q < ∞, there exists a constant K_{q} such that

(∫_{0}^{1}∥∑_{i=1}^{n} ε_{i}(t) x_{i}∥^{q} dt)^{1/q} ≤ K_{q} ∫_{0}^{1}∥∑_{i=1}^{n} ε_{i}(t) x_{i}∥ dt

for every n ≥ 1 and every family (x_{i})_{i=1}^{n} of vectors in a Banach space.

It is easy to show that when X has type or cotype, then the same holds for the space L_{2}(X) of X-valued square integrable functions. This fact is used below in section 5 and section 6.

3 Ultrapowers and some operator lemmas

In the next section about Krivine’s theorem, we use a classical fact for operators on a complex Banach space X: if λ is a boundary point of the spectrum Sp(T) of T ∈ L(X), then λ is an approximate eigenvalue for T, which means that there exists a sequence (x_{n}) ⊂ X of norm one vectors such that lim_{n} (T (x_{n}) − λx_{n}) = 0. We shall need a slightly less classical fact about commuting operators, which is very easy to obtain using the notion of ultrapower (Lemma 1 below). We shall first recall a few facts about ultrapower techniques. These techniques became popular in Banach space theory after the paper by Dacunha-Castelle and Krivine [DK]; approximately at the same time, similar objects were introduced for C^{∗}-algebras [Ja]. The limit spaces used by James [J2] in his study of super-reflexivity, the spreading models of Brunel-Sucheston [BS1], belong to the same family of tools which make possible to construct an abstract space from different pieces taken at different places.

Suppose that U is a non-trivial ultrafilter on ℕ. If X is a Banach space, we consider in X_{∞} := ℓ_{∞}(X) the closed subspace K_{U} of all sequences y = (y_{n}) ∈ X_{∞} such that lim_{n→U} ∥y_{n}∥ = 0, and we let X_{U} be the quotient space X_{∞}/K_{U}. Let π_{U} denote the quotient map from X_{∞} to X_{U}. If x = (x_{n}) and ξ = π_{U}(x), then ∥ξ∥ = lim_{n→U} ∥x_{n}∥. We have a canonical isometry i_{X,U} from X to X_{U} that sends x ∈ X to the class of the constant sequence x = (x_{n}) where x_{n} = x for every n. Using this isometric embedding we shall consider that X ⊂ X_{U}.

The crucial fact is here: suppose that η_{1}, …, η_{ℓ} ∈ X_{U} are represented by sequences y_{j} = (y_{j,n})_{n≥0} ∈ X_{∞}, for j = 1, …, ℓ, and that we have a finite number of inequality relations

(R)  a_{i} < ∥x_{i} + ∑_{j=1}^{ℓ} b_{i,j} η_{j}∥ < c_{i}, i = 1, …, k,

where a_{i}, c_{i} ∈ ℝ, x_{i} ∈ X, (b_{i,j}) is a matrix of scalars. Let us say that a property depending upon n ∈ ℕ is true when n is U-large if the set A ⊂ ℕ of those n for which the property holds belongs to U; then we can say that when n is U-large, we have in X

(R_{n})  a_{i} < ∥x_{i} + ∑_{j=1}^{ℓ} b_{i,j} y_{j,n}∥ < c_{i}, i = 1, …, k.

This implies that X_{U} is finitely representable in X (and slightly more: if E is any finite dimensional subspace of X_{U}, we can find a (1 + ε)-isomorphism T from E into X such that T (x) = x for every x ∈ E ∩ X). We see that F belongs to F(X) if and only if F is isometric to a subspace of X_{U}. Every super-property of X passes to X_{U}, for example type or cotype.

Suppose now that T is a bounded linear operator on X. We define T_{∞} on X_{∞} in the obvious way,

T_{∞}(x) = (T (x_{n})),

whenever x = (x_{n}) ∈ X_{∞}. It is clear that K_{U} is stable under T_{∞}, so that T_{∞} induces a bounded linear map T_{U} on X_{U}. It is easy to check that T → T_{U} is an isometric homomorphism of unital Banach algebras from L(X) to L(X_{U}).

Using the above principle (R) ⇒ (R_{n}), we see that if x = (x_{n}) ∈ X_{∞} and if ξ = π(x) ∈ X_{U}, then this vector ξ satisfies T_{U}(ξ) = λξ if and only if lim_{n→U} (T (x_{n}) − λx_{n}) = 0; in particular, if X is complex, for every boundary point λ of the spectrum of T we can find a sequence (x_{n}) ⊂ X of norm one vectors such that lim_{n} (T (x_{n}) − λx_{n}) = 0, which shows that the eigenspace ker(T_{U} − λI) is not trivial.

Lemma 1 Suppose that X is a complex Banach space, and that S, T are commuting bounded linear operators on X. If (x_{n}) ⊂ X is a sequence of norm one vectors such that T (x_{n}) − λx_{n} tends to 0, we can find μ ∈ ℂ and a norm one vector x ∈ X such that T (x) ∼ λx and S(x) ∼ μx.

PROOF. We know that X_{λ} = ker(T_{U} − λI) is not {0}, and S_{U} commutes with T_{U}, therefore X_{λ} is stable under S_{U}. If μ is a boundary point of the spectrum of the restriction of S_{U} to X_{λ}, we can find a norm one vector ξ in X_{λ} such that S_{U}(ξ) ∼ μξ. Bringing back ξ to X — using (R) ⇒ (R_{n}), with η_{1} = ξ, η_{2} = T_{U}(ξ) and η_{3} = S_{U}(ξ) — we obtain for every ε > 0 a norm one vector x ∈ X such that ∥T (x) − λx∥ < ε and ∥S(x) − μx∥ < ε.

Let X be a complex Banach space, and let T be an into isomorphism from X into X, with ∥x∥ ≤ C ∥T (x)∥ for every x ∈ X. For every integer n ≥ 1, we may define K_{n} as the smallest constant for which

∥x∥ ≤ K_{n} ∥T^{n}(x)∥

for every x ∈ X. It is clear that K_{m+n} ≤ K_{m} K_{n}, so that r = lim_{n} K_{n}^{1/n} exists by a standard lemma. Also, K_{n} ≤ C^{n} and K_{n} ∥T^{n}(x)∥ ≤ K_{n} ∥T ∥^{n}∥x∥ yield that 0 < ∥T ∥^{−1} ≤ r ≤ C.

Lemma 2 There exists λ ∈ ℂ with |λ| = r and a sequence (x_{n}) of norm one vectors in X such that lim_{n} (T (x_{n}) − λ^{−1}x_{n}) = 0.

PROOF. We introduce an operator S of which r will be the spectral radius; this S acts as a sort of inverse for T_{U}. For every x ∈ X, let N(x) denote the supremum of k such that x belongs to the range of T^{k} (this value N(x) may be +∞). Let Z_{0} be the subspace of X_{U} consisting of all ξ that have a representative x = (x_{n}) such that lim_{U} N(x_{n}) = +∞. It is obvious that Z_{0} is stable under T_{U}; let Z be the closure in X_{U} of Z_{0}, and let T_{Z} denote the restriction of T_{U} to Z.

When ξ ∈ Z_{0}, we see that ξ = T_{Z}(η) for some (unique) η: indeed, if x = (x_{n}) belongs to the class of ξ and lim_{U} N(x_{n}) = +∞, we have that N(x_{n}) ≥ 1 when n is U-large, which means that A = {n : N(x_{n}) ≥ 1} ∈ U; hence for every n ∈ A we have x_{n} = T (y_{n}) for some y_{n} ∈ X; if we let y_{n} = 0 for n ∉ A, then y = (y_{n}) satisfies lim_{n→U} N(y_{n}) = +∞ (because N(y_{n}) ≥ N(x_{n}) − 1); if η = π(y), then η belongs to Z and T_{Z}(η) = ξ; clearly ∥η∥ ≤ C ∥ξ∥. This shows that T_{Z} is invertible in L(Z).

Let S = T_{Z}^{−1}. It is quite clear that ∥S^{n}∥ ≤ K_{n}, so that the spectral radius ρ(S) = lim_{k} ∥S^{k}∥^{1/k} of S satisfies ρ(S) ≤ r; we shall see that r = ρ(S). Let us fix k ≥ 2 and ε > 0. For n large, we know that K_{nk} > (r − ε)^{nk}, thus we can find a vector x_{n} ∈ X such that ∥x_{n}∥ > (r − ε)^{nk}∥T^{nk}(x_{n})∥. Let h be a large integer, but small compared to n, say h − 1 < \sqrt{n} ≤ h for example. If we had

∥T^{jk}(x_{n})∥ ≤ (r − 2ε)^{k} ∥T^{jk+k}(x_{n})∥ = (r − 2ε)^{k} ∥T^{k}(T^{jk}(x_{n}))∥

for every j = h, …, n − 1, it would follow that

(r − ε)^{nk}∥T^{nk}(x_{n})∥ < ∥x_{n}∥ ≤ C^{hk} (r − 2ε)^{nk−hk} ∥T^{nk}(x_{n})∥,

which is impossible when n is large. For every n ≥ n_{0}, and for some j such that \sqrt{n} ≤ j < n, we may thus find a vector y_{n} = α T^{jk}(x_{n}) such that 1 = ∥y_{n}∥ > (r − 2ε)^{k} ∥T^{k}(y_{n})∥, and this vector satisfies N(y_{n}) ≥ k\sqrt{n}. If y = (y_{n}) and η = π(y) we get η ∈ Z and ∥S^{k}(η)∥ > (r − 2ε)^{k} ∥η∥. It follows that the spectral radius of S is larger than r − 2ε, hence equal to r.

Let λ ∈ Sp(S) be such that |λ| = r. It follows from the “boundary of the spectrum lemma” that we can find a norm one vector ξ ∈ Z such that S(ξ) ∼ λ ξ, or T_{Z}(ξ) ∼ λ^{−1}ξ; bringing back ξ to X in the usual way gives a norm one vector x for which T (x) ∼ λ^{−1}x, as was to be proved.

4 Krivine’s theorem

See [MS, Chapter 12] or [BLi, Chapter 12] for a more precise presentation of the results of this section. I prefer here to tell a pleasant story, rather than being too technical. Roughly speaking, Krivine’s theorem says that every Banach space X contains (1 + ε)-isomorphs of ℓ_{p}^{n}, for some p ∈ [1, +∞] and every n ≥ 1, or in other words it says that some ℓ_{p} (or c_{0}, when p = +∞) is finitely representable in X. More precise statements tell us that, given a basic sequence in X, or simply a sequence (x_{n}) with no Cauchy subsequence, there exists p ∈ [1, +∞] such that for every n ≥ 1 and ε > 0, we can find blocks of the given sequence that are (1 + ε)-equivalent to the unit vector basis of ℓ_{p}^{n}. It is sometimes useful to be more specific, and to predict what values of p can be realized, starting from some norm invariants of the sequence (x_{n}). This will be the case in the next section about type, cotype and the MP+K theorem.

The proofs of Krivine’s theorem are usually divided into two steps: the first step replaces the given sequence by one that has some minimal regularity; this step uses only subsequences, or just differences of two vectors from the original sequence (as opposed to the second step, that requires clever long blockings). The argument is due to Brunel and Sucheston: given a sequence with no Cauchy subsequence, and using Ramsey’s theorem, we may find a subsequence which is asymptotically invariant under spreading, see [BS1], and also [Go]; alternatively, this can be achieved by general abstract arguments involving iterated ultrapowers, usual in model theory where a somewhat parent notion of indiscernible sequence is defined. Given a Banach space X and a space Y of scalar sequences, we say that Y is a spreading model for X if there exists a normalized sequence (x_{n}) ⊂ X, with no Cauchy subsequence, such that

∥ ∑_{j=1}^{k} a_{j} e_{j}∥_{Y} = lim ∥ ∑_{j=1}^{k} a_{j} x_{n_{j}}∥_{X}

for every k ≥ 1 and all scalars (a_{j})_{j=1}^{k}; the limit is taken when n_{1} → ∞ and n_{1} < n_{2} < … < n_{k}, and (e_{j}) denotes the standard unit vector basis for the space of scalar sequences.

The second part of this first step, also due to Brunel and Sucheston, is to observe that the differences (e_{2j+1} − e_{2j}) are suppression-unconditional in Y (see below for a definition); further, the differences are bounded away from zero because the sequence (x_{n}) had no Cauchy subsequence; this implies that we can find 2-unconditional finite sequences (z_{i})_{i=1}^{k} in X, with k as large as we wish, whose vectors z_{i} are differences z_{i} = x_{n_{2i}} − x_{n_{2i−1}} of two suitable vectors from the given sequence (x_{n}). The spreading model Y is finitely representable in X, in a special way: any finite sequence (y_{k}) of blocks of the basis in Y can be sent to blocks from the sequence (x_{n}) in X. We shall therefore present the rest of the proof of Krivine’s theorem assuming that we start from this situation, replacing the original space X by a spreading model X′, which is (block) finitely representable in X and has a nice basis. The real thing is to prove Krivine’s theorem for X′.

Let X be a Banach space with a basis (e_{n})_{n≥0}; we say that this basis is a suppression-unconditional basis when for every x ∈ X, the norm does not increase if we replace one of the non-zero coordinates of x by 0; this yields that the basis is unconditional, with unconditionality constant ≤ 2 (in the real case). Let X be a Banach space with a suppression-unconditional basis (e_{n})_{n≥0}; we say that the norm is invariant under spreading if for every integer k ≥ 0 and all n_{0} < n_{1} < … < n_{k},

∥∑_{j=0}^{k} a_{j} e_{n_{j}}∥ = ∥∑_{j=0}^{k} a_{j} e_{j}∥

for all scalars (a_{j}). Let x = ∑_{j=0}^{k} a_{j} e_{j} be a vector with finite support in X; we say that y is a copy of x if y = ∑_{j=0}^{k} a_{j} e_{n_{j}} for some n_{0} < n_{1} < … < n_{k}. If x = ∑ a_{j}e_{j} and y = ∑ b_{j}e_{j}, we write x < y when all non-zero coordinates of x appear before those of y, that is max{j : a_{j} ≠ 0} < min{j : b_{j} ≠ 0}. We say that x_{1}, …, x_{n} are successive vectors if x_{1} < x_{2} < … < x_{n}.

After the preliminary work has been done, the heart of Krivine’s result is the following Theorem 3. The arguments of Brunel-Sucheston imply that for every Banach space X, we can find a space X_{0} with a suppression-unconditional basis, invariant under spreading, such that X_{0} is finitely representable in X; if X_{0} contains ℓ_{p}^{k}, then X will also. We shall therefore assume that X is a Banach space with a suppression-unconditional basis (e_{n})_{n≥0}, and a norm invariant under spreading. For every integer n ≥ 1, let R_{n} be the smallest constant and S_{n} be the largest constant such that for every x ∈ X, we have

S_{n} ∥x∥ ≤ ∥∑_{i=1}^{n} x_{i}∥ ≤ R_{n} ∥x∥

whenever x_{1} < x_{2} < … < x_{n} are successive copies of x.

Theorem 3 Let X be a Banach space with a suppression-unconditional basis (e_{n})_{n≥0}, and a norm invariant under spreading; suppose that p ≥ 1 is defined by the equation

(a) : 2^{1/p} = lim sup_{n}(R_{2^{n}})^{1/n} or (b) : 2^{1/p} = lim inf_{n}(S_{2^{n}})^{1/n}.

For every k ≥ 1 and ε > 0 it is possible to find k successive blocks x_{1} < … < x_{k} in X that are (1 + ε)-equivalent to the unit vector basis of ℓ_{p}^{k}, and that are copies of some norm one vector x ∈ X.

PROOF. Let I be the set of rational numbers r such that 0 ≤ r < 1, let (f_{r})_{r∈I} be the standard unit vector basis for ℝ^{(I)}, and let us define a norm on the linear span Y_{0} of (f_{r})_{r∈I} as follows: if r_{0} < r_{1} < … < r_{k}, let

∥∑_{j=0}^{k} a_{j} f_{r_{j}}∥_{Y} = ∥∑_{j=0}^{k} a_{j} e_{j}∥_{X}

for all scalar coefficients (a_{j}). If Y_{0} is real, we complexify it in any reasonable way, for example

∥x + iy∥ = sup_{θ} ∥ sin(θ) x + cos(θ) y∥,

which preserves invariance under spreading and unconditionality. Let Y be the completion of Y_{0}; it is clear that (f_{r}) is a suppression-unconditional basis for Y, invariant under spreading in the new context. We say that y′ ∈ Y is a copy of y = ∑_{r∈I} a_{r} f_{r} if y′ = ∑_{r∈I} a_{r} f_{ϕ(r)} for some increasing map ϕ from I into itself. What we mean by successive copies of a given vector in Y is clear. It is also clear that Y is finitely representable in X, and a finitedimensional subspace of Y generated by successive copies of some vector in Y can be approximated by a subspace of X, generated by successive copies of some vector in X.

We can now relate the behaviour of sums of copies of vectors in X to the properties of some linear operators defined on this space Y. Indeed, we may define a doubling operator D on Y by the formula

∀y ∈ Y, D(y) = ∑_{0≤r<1/2} y(2r) f_{r} + ∑_{1/2≤r<1} y(2r − 1) f_{r},

or D(y)(r) = y(2r mod 1), considering y as a function I → ℂ. For every y ∈ Y_{0}, the vector D(y) is the sum of two copies y_{1} < y_{2} of y, hence ∥y∥ ≤ ∥D(y)∥ ≤ 2 ∥y∥. It is clear that the constant R_{2^{n}} for the initial space X is equal to the norm of D^{n}, therefore in case (a), we see that 2^{1/p} is the spectral radius of D. We may thus find λ ∈ ℂ with |λ| = 2^{1/p} and a norm one vector z ∈ Y_{0} such that D(z) ∼ λz. In case (b), the constant S_{2^{n}} appears to be the reciprocal of the constant K_{n} associated to the into isomorphism D (see before Lemma 2), therefore if 2^{1/p} = lim_{n} S_{2^{n}}^{1/n}, we know by Lemma 2 that we can again find λ ∈ ℂ with |λ| = 2^{1/p} and a norm one vector z ∈ Y_{0} such that D(z) ∼ λz. Using unconditionality, we get D(|z|) ∼ |λ| |z|. In both cases (a) and (b) we found a norm one vector y = α|z| ∈ Y_{0} (with 1/2 ≤ α ≤ 2) such that D(y) ∼ 2^{1/p} y. Reproducing y in X gives a norm one vector x ∈ X such that, when x_{1} < x_{2} are copies of x, then x_{1} + x_{2} is very close to some copy x′ of 2^{1/p} x. I like to call such a vector x a Krivine vector. Suppose that x_{1} < x_{2} < … < x_{k} are copies of this vector x. If n ≥ 1 is given and if D(y) − 2^{1/p} y has norm smaller than some ε_{n} > 0, we deduce that

(K)  ∥∑_{j=1}^{k} a_{j}x_{j}∥^{p} ∼ ∑_{j=1}^{k} a_{j}^{p},

provided all coefficients are of the form a_{j} = 2^{−k_{j}/p}, for some integer k_{j} such that 0 ≤ k_{j} ≤ n, and ∑_{j} a_{j}^{p} = 1 (if K = max k_{j}, replace each a_{j}x_{j} by 2^{K−k_{j}} copies of 2^{−K/p} x; this gives 2^{K} copies of 2^{−K/p} x, which we may group two by two again, obtaining after K steps a single copy of the vector x).

This is not quite enough, and we also introduce an operator T on Y which reproduces three times every vector y ∈ Y,

T (y) = ∑_{0≤r<1/3} y(3r) f_{r} + ∑_{1/3≤r<2/3} y(3r − 1) f_{r} + ∑_{2/3≤r<1} y(3r − 2) f_{r}.

It is clear that DT = T D is the operator that replaces every vector x by six copies of x; the commutation property and Lemma 1 enable us to find a norm one vector z such that D(z) ∼ 2^{1/p}z and T (z) ∼ μz; then T (|z|) ∼ |μ| |z|, so that we may assume that z and μ are real and ≥ 0. Some simple lattice arguments (involving comparisons of the norms of sums of respectively 2^{h}, 3^{i} and 2^{j} copies of z when 2^{h} < 3^{i} < 2^{j}) show that necessarily μ = 3^{1/p}.

If D(z) − 2^{1/p} z and T (z) − 3^{1/p} z are small enough, and if z_{1} < z_{2} < … < z_{k} are copies of this vector z, we may try to extend relation (K) to coefficients (a_{j}) such that a_{j} = 2^{ℓ_{j}}3^{m_{j}} for some ℓ_{j}, m_{j} ∈ ℤ; since these values are dense in [0, ∞), we are in a good position. However, dealing with the error terms is painful, and we may instead pass to the ultrapower Y_{U}, which is still a lattice, with a linear ordering defined in this way: we say that ξ < η if ξ and η have representatives (x_{n}) and (y_{n}) with x_{n} < y_{n} for every n, and we say that η is a copy of ξ if ξ and η have representatives (x_{n}) and (y_{n}) such that y_{n} is a copy of x_{n} for every n; in Y_{U} we can find a norm one vector η such that D_{U}(η) = 2^{1/p}η and T_{U}(η) = 3^{1/p}η; to get this, we take for η the class of a normalized sequence (z_{n}) in Y with D(z_{n}) − 2^{1/p}z_{n} → 0 and T (z_{n}) − 3^{1/p}z_{n} → 0. In this framework where we have equalities, it is easy to prove that when η_{1}, …, η_{k} are successive copies of this vector η and when the coefficients (a_{j}) satisfy a_{j} = 2^{ℓ_{j}}3^{m_{j}}, with ℓ_{j}, m_{j} ∈ ℤ, then ∥ ∑_{j=1}^{k} a_{j}η_{j}∥^{p} = ∑_{j=1}^{k} a_{j}^{p}; next, we extend this by density to all non-negative scalars. Going back to X, and using the special form of the vectors η_{1}, …, η_{k}, we can find successive copies x_{1}, …, x_{k} of some norm one vector x in X such that

(K′)  (1 + ε)^{−p/2} ∑_{j=1}^{k} a_{j}^{p} ≤ ∥∑_{j=1}^{k} a_{j}x_{j}∥^{p} ≤ (1 + ε)^{p/2} ∑_{j=1}^{k} a_{j}^{p},

for all non-negative scalars (a_{j}). Everything would be fine if the basis in X was 1-unconditional, but it is not so: what we get so far is a sequence x_{1}, …, x_{k} which is 2(1 + ε)-equivalent to the ℓ_{p}^{k}-basis in the real case, and 4(1 + ε) in the complex case, for every k ≥ 2: if v = ∑ a_{j}x_{j}, the ℓ_{p}-norm of the coefficients is dominated by (∥v^{+}∥^{p} + ∥v^{−}∥^{p})^{1/p} ≤ 2^{1/p} ∥v∥, using first (K′) then suppression unconditionality; in the other direction use ∥v∥ ≤ 2^{1−1/p} (∥v^{+}∥^{p} + ∥v^{−}∥^{p})^{1/p}.

Suppose p < ∞ for simplicity; if k = m^{2} and if we form new blocks y_{1}, …, y_{m} in X of the form y_{i} = m^{−1/p} ∑_{j=1}^{m} (−1)^{j}x_{m(i−1)+j}, then (y_{1}, …, y_{m}) is still a sequence of successive copies of some y ∈ X, hence invariant under spreading, 5-equivalent to the ℓ_{p}^{m} basis (say), but the unconditional constant is improved to something arbitrarily close to 1 as m grows (in the complex case, a similar trick using a primitive root of unity does the required job). We build a limit space X_{1} from the sequence ([y_{1}^{(m)}, …, y_{m}^{(m)}])_{m}, by setting

∥∑_{j=1}^{n} a_{j} e_{j}∥_{X_{1}} = lim_{m→U} ∥ ∑_{j=1}^{n∧m} a_{j} y_{j}^{(m)}∥

for every n ≥ 1 and all scalars (a_{j}). This space X_{1} is finitely representable in X, with a 1-unconditional basis, invariant under spreading, and 5-equivalent to the ℓ_{p} basis. In X_{1} we have clearly 2^{1/p} = lim sup_{n}(R_{2^{n}})^{1/n}. Applying the above construction to X_{1} gives new blocks x_{1}, …, x_{k} that satisfy (K′) in X_{1}: this finishes the proof, since the basis in X_{1} is 1-unconditional.

The proof above is due to Lemberg [Le], who was Krivine’s PhD student in the years ’80. The fundamental facts are still the same as in the original paper [Kr2], but the details in [Kr2] are harder to follow. Combining the arguments of Brunel-Sucheston and the preceding Theorem, we obtain one of the usual forms of Krivine’s theorem.

Corollary 4 Suppose that X is a real or complex Banach space, and (x_{n}) a bounded sequence in X with no Cauchy subsequence. For some p ∈ [1, +∞], for every k ≥ 1 and ε > 0 it is possible to find k successive blocks of the given sequence that are (1 + ε)-equivalent to the unit vector basis of ℓ_{p}^{k}.

Our next Corollary is expressed in a slightly unnatural way, but suitable for the next section.

Corollary 5 Suppose r, s ≥ 1 are given. If for some κ > 0 and for every n ≥ 2, a Banach space X contains a normalized suppression-unconditional sequence y^{(n)} = (y_{1}^{(n)}, …, y_{n}^{(n)}) such that

∥∑_{i∈C} y_{i}^{(n)}∥ ≥ κ |C|^{1/r}

for every subset C ⊂ {1, …, n}, or such that

∥∑_{i∈C} y_{i}^{(n)}∥ ≤ κ |C|^{1/s}

for every subset C ⊂ {1, …, n}, then for some p ≤ r (or p ≥ s) and for every k ≥ 1, ε > 0, it is possible when n ≥ N(k, ε) to form k successive blocks of the given sequence y^{(n)} that are (1 + ε)-equivalent to the unit vector basis of ℓ_{p}^{k}.

PROOF. We construct as we did before a limit space X′ from the long sequences as follows. Using Brunel-Sucheston principle, we may select from our long sequences (y_{i}^{(n)}) some (finite) subsequences z_{1}^{(n)}, …, z_{k_{n}}^{(n)} that are almost indiscernible, and have a length k_{n} tending to ∞ with n; then we define a norm on c_{00} (the space of finitely supported scalar sequences) by

∥ ∑_{i=1}^{m} c_{i}e_{i}∥_{X′} = lim_{n→U} ∥ ∑_{i=1}^{m∧k_{n}} c_{i}z_{i}^{(n)}∥

where (e_{i})_{i≥0} denotes the unit vector basis of c_{00}. Notice that when n is Ularge, the length k_{n} exceeds m; this yields that (e_{i}) is normalized in X′. We obtain a space X′ with a normalized suppression-unconditional basis and a norm invariant under spreading. In the first case, we get for every n ≥ 1

κ n^{1/r} ≤ ∥∑_{i=1}^{n} e_{i}∥_{X′} ≤ R_{n} ∥e_{1}∥_{X′} = R_{n},

and similarly in the second case we obtain that S_{n} ≤ κ n^{1/s}. We know from Theorem 3 that we may get ℓ_{p}^{k} in X′, with p such that 2^{1/p} = lim_{n} (R_{2^{n}})^{1/n} or 2^{1/p} = lim_{n} (S_{2^{n}})^{1/n}, thus p ≤ r in the first case and p ≥ s in the second.

5 Type, cotype and ℓ_{p}^{n}s. The MP+K theorem

Let X be a Banach space. We denote by p_{X} the supremum of all p such that X has type p, and by q_{X} we denote the infimum of all q such that X has cotype q. It is clear using Khintchine’s inequality that p_{X} ≤ 2 ≤ q_{X}, already when X = ℝ.

Theorem 6 Let X be an infinite dimensional Banach space; for every integer k ≥ 1 and ε > 0, the space X contains (1 + ε)-isomorphs of ℓ_{p_{X}}^{k} and of ℓ_{q_{X}}^{k}.

For the type case and 1 < p < 2, there exists a quantitative estimate due to Pisier [P7], see also [MS, Theorem 13.12]. The dimension k of a good isomorph in X of ℓ_{p}^{k} is given there as a function of the stable type p constant ST_{p}(X) of the normed space X.

PROOF. If p_{X} = 2 we may use Dvoretzky’s theorem [D2]. Assume p_{X} < 2 and choose r such that p_{X} < r < 2. For each n ≥ 1, let φ(n) denote the smallest constant such that

∫_{0}^{1}∥∑_{i=1}^{n} ε_{i}(t)x_{i}∥^{r} dt ≤ φ(n)^{r} ∑_{i=1}^{n} ∥x_{i}∥^{r}

for every family x_{1}, …, x_{n} of n vectors in X. It is clear that φ is non-decreasing, and tends to +∞ since X does not have type r. Suppose that x_{1}, …, x_{n} are chosen in X so that ∑_{i=1}^{n} ∥x_{i}∥^{r} = 1 and

∫_{0}^{1}∥∑_{i=1}^{n} ε_{i}(t)x_{i}∥^{r} dt > \frac{1999}{2000} φ(n)^{r}.

We shall use an exhaustion argument inspired by Nikišin’s paper [N2]. Let (B_{α})_{α∈I} be a maximal family of disjoint subsets of {1, …, n} such that

∫_{0}^{1}∥ ∑_{i∈B_{α}} ε_{i}(t)x_{i}∥^{r} dt < \frac{1}{2000} ∑_{i∈B_{α}} ∥x_{i}∥^{r}.

If B denotes the union of these sets B_{α}, and m denotes the cardinality of I (notice that m < n because |B_{α}| > 1), we get

∫_{0}^{1}∥∑_{i∈B} ε_{i}(t)x_{i}∥^{r} dt = ∫ ∥∑_{α∈I} ε_{α}(s)( ∑_{i∈B_{α}} ε_{i}(t)x_{i})∥^{r} ds dt

≤ φ(m)^{r} ∑_{α∈I} ∫_{0}^{1}∥ ∑_{i∈B_{α}} ε_{i}(t)x_{i}∥^{r} dt ≤ \frac{φ(m)^{r}}{2000} ∑_{α∈I} ∑_{i∈B_{α}} ∥x_{i}∥^{r}

≤ \frac{φ(n)^{r}}{2000} ∑_{i=1}^{n} ∥x_{i}∥^{r} = \frac{φ(n)^{r}}{2000} .

Let A denote the complement of B and for every j ≥ 0 let

A_{j} = {k ∈ A : 2^{−j−1} < ∥x_{k}∥ ≤ 2^{−j}}.

Observe that ∥x_{k}∥ ≤ 1 for every k because ∑_{i=1}^{r} ∥x_{i}∥^{r} = 1, so that the sets (A_{j})_{j≥0} cover the set A. Let N = max_{j} |A_{j}| denote the maximal cardinality of the sets (A_{j})_{j≥0}. Then

(∫_{0}^{1}∥∑_{i∈A} ε_{i}(t)x_{i}∥^{r} dt)^{1/r} ≤ ∑_{j=0}^{+∞}(∫_{0}^{1}∥ ∑_{i∈A_{j}} ε_{i}(t)x_{i}∥^{r} dt)^{1/r} ≤ N ∑_{j=0}^{+∞} 2^{−j} = 2 N.

We obtain

(\frac{1999}{2000})^{1/r} φ(n) < (∫_{0}^{1}∥∑_{i=1}^{n} ε_{i}(t)x_{i}∥^{r} dt)^{1/r} ≤ \frac{φ(n)}{2000^{1/r}} + 2 N

which shows that N is big when φ(n) is big. Let j_{0} be such that |A_{j_{0}}| = N. By maximality of B we obtain for every non-empty subset C of A_{j_{0}}

∫_{0}^{1}∥∑_{i∈C} ε_{i}(t)x_{i}∥^{r} dt ≥ \frac{1}{2000} ∑_{i∈C} ∥x_{i}∥^{r} ≥ \frac{2^{−(j_{0}+1)r}}{2000} |C|.

Replacing the vectors (x_{i})_{i∈A_{j_{0}}} by normalized vectors (y_{i}), we obtain a normalized sequence (y_{1}, y_{2}, …, y_{m}), as long as we wish, such that

(∫_{0}^{1}∥∑_{i∈C} ε_{i}(t) y_{i}∥^{r} dt)^{1/r} ≥ κ |C|^{1/r}

for every subset C of \{1, …, m\} (with κ = \frac{1}{2} 2000^{−1/r}). This inequality remains true if we replace the L_{r}(X) norm by the norm of L_{1}(X) and κ by some κ′ > 0 (use Kahane’s inequalities). For every n ≥ 1, we may thus find an unconditional normalized sequence in L_{1}(X), of the form (ε_{j}y_{j}^{(n)})_{j=1}^{n}, with the above property, and since r < 2 it implies that for some c = c(r, κ′) > 0, we have ∥ ∑_{j=1}^{n} a_{j}ε_{j}y_{j}^{(n)}∥_{L_{1}(X)} ≥ c (∑_{j=1}^{n} |a_{j}|^{2})^{1/2} for all scalars. From Corollary 5 follows that for every integer m, we can, when n is large enough, get blocks z_{1}, …, z_{m} ∈ L_{1}(X) of (ε_{j}y_{j}^{(n)})_{j=1}^{n} that are (1 + ε)-equivalent to the unit vector basis of ℓ_{p}^{m} for some p ≤ r, and the ℓ_{2}-norm of the coefficients in each block z_{i} is bounded by c(r, κ′)^{−1}. By Kahane’s inequalities again, all L_{s}(X) norms are equivalent on the span of (ε_{j}y_{j}^{(n)})_{j=1}^{n}, hence the sequence (z_{1}, …, z_{m}) considered in L_{2}(X) is uniformly equivalent to the unit vector basis of ℓ_{p}^{m}; since L_{2}(X) has type s whenever X has type s, we have for every s < p_{X} and for some constants K, K_{s}

K^{−1}m^{1/p} ≤ ∫_{0}^{1} ∥ ∑_{i=1}^{m} ε_{i}(t)z_{i}∥_{L_{2}(X)} dt ≤ K_{s} m^{1/s}

for every m ≥ 1. This yields that s ≤ p, for every s < p_{X}, hence p_{X} ≤ p. Starting with a long enough sequence (z_{i})_{i=1}^{m} and blocking again in the ℓ_{p}- sense we may find three blocks b_{1}, b_{2}, b_{3} ∈ L_{1}(X) of some sequence (ε_{j}y_{j}), supported on three disjoint intervals J_{1}, J_{2}, J_{3} and such that, letting ω = (ε_{j}) and

b_{i}(ω) = ∑_{j∈J_{i}} a_{j} ε_{j} y_{j}, i = 1, 2, 3,

then the three functions b_{1}, b_{2}, b_{3} are (1 + ε)-equivalent to the unit vector basis of ℓ_{p}^{3} in the norm of L_{1}(X), and the coefficients satisfy ∑_{j∈J_{i}} |a_{j}|^{2} < τ^{2}/12 for i = 1, 2, 3 and a small τ > 0 (use p < 2). For every fixed triple (c_{1}, c_{2}, c_{3}) of scalars, this implies by Azuma’s inequality (see [MS, 7.4]) a strong concentration for the set of ω such that

∥c_{1} b_{1}(ω) + c_{2} b_{2}(ω) + c_{3} b_{3}(ω)∥ ∼ (|c_{1}|^{p} + |c_{2}|^{p} + |c_{3}|^{p})^{1/p},

and allows us to select a choice of ω = (ε_{j}) that works for all (c_{i})_{i=1}^{3}, by a standard δ-net argument on the unit sphere of ℓ_{p}^{3}; this shows that for most of the choices ω of signs, the vectors b_{1}(ω), b_{2}(ω), b_{3}(ω) in X form a nice copy of the unit vector basis of ℓ_{p}^{3}. We may choose r close enough to p_{X} so that ℓ_{p_{X}}^{3} is almost isometric to ℓ_{p}^{3}, since p_{X} ≤ p ≤ r, and this ends the proof in this case k = 3. The reader will easily pass from 3 to an arbitrary integer k.

Let us be more specific about the use of Azuma’s inequality. On the space Ω = {−1, 1}^{n}, we define for every c in the unit sphere of ℓ_{p}^{3} the function

f_{c}(ω) = f_{c}(ε_{1}, …, ε_{n}) = ∥∑_{i=1}^{3} c_{i} ( ∑_{j∈J_{i}} a_{j}ε_{j} y_{j})∥

and we consider the finite martingale

M_{j}(ε_{1}, …, ε_{j}) = ∫ f_{c}(ε_{1}, …, ε_{n}) dε_{j+1} … dε_{n},

for j = 0, …, n. The differences (d_{j})_{j=0}^{n} of this martingale satisfy |d_{j+1}| = |M_{j+1} − M_{j}| ≤ |a_{j+1}|, hence S^{2} = ∑_{j=1}^{n} |d_{j}|^{2} ≤ τ^{2}/4. Azuma’s inequality gives

P ({ω ∈ Ω : |f_{c}(ω) − M_{0}| ≥ t}) ≤ 2 exp(−t^{2}/(4S^{2})) ≤ 2 exp(−t^{2}/τ^{2})

for every t > 0, where M_{0} = M_{0}(c) is equal to the norm of c_{1}b_{1} + c_{2}b_{2} + c_{3}b_{3} in L_{1}(X), which is (1 + ε)-equivalent to the ℓ_{p}^{3}-norm of c, namely 1. If Λ is a δ-net on the unit sphere of ℓ_{p}^{3} and if τ was so small that 2 |Λ| < exp(δ^{2}/τ^{2}), we may find ω such that

|∥c_{1}b_{1}(ω) + c_{2}b_{2}(ω) + c_{3}b_{3}(ω)∥ − M_{0}(c)| ≤ δ

for every c ∈ Λ, from which the result follows.

Let us pass to the cotype case. If q_{X} = 2 we may use Dvoretzky’s theorem. Assume q_{X} > 2. We choose s such that q_{X} > s > 2. Let ψ(n) denote the smallest constant such that

∑_{i=1}^{n} ∥x_{i}∥^{s} ≤ ψ(n)^{s} ∫_{0}^{1}∥∑_{i=1}^{n} ε_{i}(t)x_{i}∥^{s} dt

for every family x_{1}, …, x_{n} of n vectors in X. It is clear that ψ is non-decreasing, and tends to +∞ since X does not have cotype s. Suppose that x_{1}, …, x_{n} are chosen in X so that ∑_{i=1}^{n} ∥x_{i}∥^{s} = 1 and

1 > \frac{1999}{2000} ψ(n)^{s} ∫_{0}^{1}∥∑_{i=1}^{n} ε_{i}(t)x_{i}∥^{s} dt.

Let (B_{α})_{α∈I} be a maximal family of mutually disjoint non-empty subsets of {1, …, n} such that

∑_{i∈B_{α}} ∥x_{i}∥^{s} ≤ \frac{1}{2000} ∫_{0}^{1}∥ ∑_{i∈B_{α}} ε_{i}(t)x_{i}∥^{s} dt.

If B denotes the union of these sets B_{α}, and m < n denotes the cardinality of the index set I, we get

∑_{i∈B} ∥x_{i}∥^{s} = ∑_{α∈I} ∑_{i∈B_{α}} ∥x_{i}∥^{s} ≤ ∑_{α∈I} \frac{1}{2000} ∫_{0}^{1}∥ ∑_{i∈B_{α}} ε_{i}(t)x_{i}∥^{s} dt

≤ \frac{ψ(m)^{s}}{2000} ∫ ∥∑_{α∈I} ε_{α}(s)( ∑_{i∈B_{α}} ε_{i}(t)x_{i})∥^{s} ds dt

≤ \frac{ψ(n)^{s}}{2000} ∫_{0}^{1}∥∑_{i=1}^{n} ε_{i}(t)x_{i}∥^{s} dt.

Let A denote the complement of B and for every j ≥ 0 let

A_{j} = {k ∈ A : 2^{−j−1} < ∥x_{k}∥ ≤ 2^{−j}}.

We have

∑_{i∈A} ∥x_{i}∥^{s} > \frac{1998}{2000} ψ(n)^{s} ∫_{0}^{1}∥∑_{i=1}^{n} ε_{i}(t)x_{i}∥^{s} dt.

Let j_{1} be the smallest j ≥ 0 such that A_{j} is not empty. If N = |A_{j_{0}}| is the largest cardinality of the sets A_{j}, then

N ∑_{j=j_{1}}^{+∞} 2^{−js} ≥ \frac{1998}{2000} ψ(n)^{s} 2^{−j_{1}s−s}

which shows that N is large when ψ(n) is large. By maximality of B,

∑_{i∈C} ∥x_{i}∥^{s} > \frac{1}{2000} ∫_{0}^{1}∥∑_{i∈C} ε_{i}(t)x_{i}∥^{s} dt

for every non-empty subset C ⊂ A_{j_{0}}. We change the (x_{i})_{i∈A_{j_{0}}} to normalized vectors, and go to a limit space X′, finitely representable in X and containing a normalized sequence (y_{i})_{i≥0} such that for some κ_{0},

κ_{0} |C|^{1/s} ≥ (∫_{0}^{1}∥∑_{i∈C} ε_{i}(t)y_{i}∥^{s} dt)^{1/s}

for every finite subset C. But this sequence (y_{i})_{i≥0} can’t have any Cauchy subsequence, or else the above property would be true with y_{i} ∼ y, in other words, true in a one dimensional setting; in this case, Khintchine’s inequality tells us that the integral is larger than |C|^{1/2} > κ_{0} |C|^{1/s}, which is impossible when |C| is large. By Brunel-Sucheston, we can pass to differences (y_{m} −y_{n}) in order to get a suppression-unconditional sequence invariant under spreading (with a poor normalization). We have

(∫_{0}^{1}∥∑_{i∈C} ε_{i}(t)(y_{2i+1} − y_{2i})∥^{s} dt)^{1/s} ≤ 2κ_{0} |C|^{1/s}

for every finite subset C, but we may now get rid of the signs (ε_{i}(t)) since the sequence of differences is 2-unconditional. We obtain therefore in X′ a normalized suppression-unconditional sequence (x_{i}) such that

∥∑_{i∈C} x_{i}∥ ≤ κ′ |C|^{1/s}

for every finite subset C. We end by applying the second case of Corollary 5.

6 K-convexity and Pisier’s theorem

When X is a type p space, then the dual X^{∗} has cotype q for the conjugate exponent (1/p+1/q = 1); this is very easy: if (x_{i}^{∗})_{i=1}^{n} is given in X^{∗}, we can find (x_{i})_{i=1}^{n} ⊂ X such that ∑_{i=1}^{n} x_{i}^{∗}(x_{i}) > (∑_{i=1}^{n} ∥x_{i}^{∗}∥^{q})^{1/q} − ε and ∑_{i=1}^{n} ∥x_{i}∥^{p} = 1; then, by orthogonality of the functions (ε_{i})

(∑_{i=1}^{n} ∥x_{i}^{∗}∥^{q})^{1/q} − ε < ∑_{i=1}^{n} x_{i}^{∗}(x_{i}) = ∫_{0}^{1}(∑_{i=1}^{n} ε_{i}(t)x_{i}^{∗})(∑_{j=1}^{n} ε_{j}(t)x_{j}) dt

≤ (∫_{0}^{1} ∥ ∑_{j=1}^{n} ε_{j}(t)x_{j}∥^{2} dt)^{1/2}(∫_{0}^{1} ∥ ∑_{i=1}^{n} ε_{i}(t)x_{i}^{∗}∥^{2} dt)^{1/2}

≤ T_{p}(X) (∫_{0}^{1} ∥ ∑_{i=1}^{n} ε_{i}(t)x_{i}^{∗}∥^{2} dt)^{1/2},

therefore C_{q}(X^{∗}) ≤ T_{p}(X). Obviously the converse is false since ℓ_{1}, dual of c_{0}, has cotype 2, while c_{0} has no non-trivial type. However, this does not happen when X^{∗} has cotype q and non-trivial type: then, X has type p. This fact was conjectured in [MP2] (although the authors had little evidence that supported this conjecture at the time), and proved by Pisier six years later [P6]. Using local reflexivity, and since type and cotype are super-properties, the preceding claim is equivalent to saying that when a Banach space Y has non-trivial type and cotype q, then the dual Y^{∗} has type p with 1/p + 1/q = 1. This will follow from the easy Lemma 7 below and from the main result of this section, Theorem 12.

Let us consider the group G = {1, −1} and let μ denote the invariant probability measure on G^{m}, that gives measure 2^{−m} to every atom. On G^{m}, let ε_{i}, i = 1, …, m denote the ith coordinate function, ε_{i}(g_{1}, …, g_{m}) = g_{i}. If α ⊂ {1, …, m} let w_{α} = ∏_{i∈α} ε_{i}; using the standard convention, we get the constant function 1 on G^{m} when α = ∅. Let |α| denote the cardinality of the set α. This family of functions (w_{α}) is the Walsh system; it is the family of characters of the abelian group G^{m}. Every function f from G^{m} to a Banach space X can be expressed as

∀ω ∈ G^{m}, f(ω) = ∑_{α} w_{α}(ω)x_{α},

for some family (x_{α}) ⊂ X. Given a function f = ∑_{α} w_{α}x_{α}, the part of the expansion corresponding to sets α with |α| = 1 is the Rademacher projection R_{X}(f) = ∑_{|α|=1} w_{α}x_{α} of the function f (we have w_{α} = ε_{i} when α = {i}).

Lemma 7 If the Rademacher projection R_{X} is bounded on L_{2}(G^{m}, μ, X) by some constant K, uniformly in m ≥ 1, then the cotype q property of X dualizes to the type p property of X^{∗}, and

T_{p}(X^{∗}) ≤ K C_{q}(X) (1/p + 1/q = 1).

PROOF. Suppose that f ∈ L_{2}(G^{m}, X); the Rademacher projection of f is of the form R_{X}f = ∑_{i=1}^{m} ε_{i}x_{i}, where x_{i} = ∫ ε_{i}(ω)f(ω) dμ(ω). The cotype q property and the boundedness of R_{X} imply that the map f → (x_{i})_{i=1}^{m} is bounded from L_{2}(G^{m}, X) to ℓ_{q}^{m}(X). It follows that the adjoint map is bounded from ℓ_{p}^{m}(X^{∗}) to L_{2}(G^{m}, X^{∗}), and this adjoint map is the map that sends (x_{i}^{∗})_{i=1}^{m} to ∑_{i=1}^{m} ε_{i}x_{i}^{∗}. We get therefore

(∫_{0}^{1} ∥ ∑_{i=1}^{m} ε_{i}(t)x_{i}^{∗}∥^{2} dt)^{1/2} ≤ ∥R_{X}∥ C_{q}(X) (∑_{i=1}^{m} ∥x_{i}^{∗}∥^{p})^{1/p}.

Definition 8 We say that X is K-convex if there exists a constant K such that for every m ≥ 1 and every function f ∈ L_{2}(G^{m}, X), expressed as f = ∑_{α} w_{α}x_{α}, we have

∥R_{X}f∥_{L_{2}} = ∥ ∑_{|α|=1} w_{α}x_{α}∥_{L_{2}} ≤ K ∥f∥_{L_{2}},

which means that ∥R_{X}∥_{L(L_{2}(X))} ≤ K. The smallest possible constant K is the K-convexity constant of X. It is equal to the supremum of ∥R_{X}∥, when the number m of Rademacher functions tends to infinity.

When this supremum is finite, we may directly define R_{X} on the infinite product G^{ℕ}, and the K-convexity constant is the norm of R_{X} on L_{2}(G^{ℕ}, X). It is clear that K-convexity is a super-property, and it passes to the dual X^{∗} with the same constant. It follows from Kahane’s inequality that the projection is also bounded in L_{q}(X) for 2 ≤ q < +∞, and using duality we see that R_{X} is then bounded in L_{p}(X) for all p such that 1 < p < +∞.

It follows from Lemma 7 that the K-convexity constant of L_{1}(G^{m}) tends to infinity with m (because L_{1} has cotype 2 while its dual L_{∞} does not have type 2), but it is instructive to give a concrete estimate. Let ĝ = (ε̂_{1}, …, ε̂_{m}) ∈ G^{m} be fixed; the function f_{g}, equal to 2^{m} at ĝ and to 0 elsewhere, has norm one in L_{1}(G^{m}), and its expansion is

f_{g} = ∑_{α} w_{α}(ĝ) w_{α}.

It follows that the function f from G^{m} to L_{1}(G^{m}) defined by

f(g, g′) = ∑_{α} w_{α}(g) w_{α}(g′)

has norm one in L_{2}(G^{m}, L_{1}(G^{m})), but its Rademacher projection (Rf)(g) = ∑_{j=1}^{m} ε_{j}(g) ε_{j} has norm ≥ \sqrt{m/2} by Khintchine’s inequality. Observe that we get K_{X} ≥ c \sqrt{log dim X}, with X = L_{1}(G^{m}). It is known that for any Banach lattice X we have K_{X} ≤ C \sqrt{1 + log dim X} (see [P4]), and the preceding simple example shows that this result is precise for lattices. For general Banach spaces, see Theorem 13 below.

Let us describe the semi-group approach: on the multiplicative group G = {1, −1} we consider for −1 ≤ c ≤ 1 the probability measure μ_{c}^{(1)} defined by

μ_{c}^{(1)} = \frac{1 + c}{2} δ_{1} + \frac{1 − c}{2} δ_{−1},

where δ_{g} denotes the unit mass at g ∈ G. Using δ_{−1} ∗ ε_{1} = −ε_{1} we get that μ_{c}^{(1)} ∗ ε_{1} = c ε_{1}. Also μ_{b}^{(1)} ∗ μ_{c}^{(1)} = μ_{bc}^{(1)}. On G^{m} we consider the m-fold tensor product μ_{c} = μ_{c}^{(m)} of m copies of μ_{c}^{(1)}. We see that μ_{c} ∗ w_{α} = c^{|α|}w_{α} and μ_{b} ∗ μ_{c} = μ_{bc}. Given a function f ∈ L_{2}(G^{m}, X), expressed as ∑_{α} w_{α}x_{α}, we see that μ_{c} ∗ f = ∑_{α} c^{|α|}w_{α}x_{α}. Since μ_{c} is a probability measure, convolution with μ_{c} is a norm 1 operator on L_{2}(G^{m}, X), for every real or complex Banach space X and every c ∈ [−1, 1].

In order to pass to the classical semi-group setting, we shall perform the following change of variable. For t ≥ 0, let ν_{t} = μ_{e}^{−t}. We get that ν_{t}∗ν_{s} = ν_{s+t}. Given a function f = ∑_{α} w_{α}x_{α} ∈ L_{2}(G^{m}, X) we set

(W)  T_{t}f = ν_{t} ∗ f = ∑_{α} e^{−|α|t} w_{α}x_{α}

and we call (T_{t})_{t≥0} the Walsh semi-group. We noticed that each T_{t} is a contraction on L_{2}(G^{m}, X). Let P_{i}, i = 1, …, m denote the projection on L_{2}(G^{m}, X) defined by

(P_{i}f)(ε_{1}, …, ε_{m}) = ∫ f(ε_{1}, …, ε_{i−1}, ε, ε_{i+1}, …, ε_{m}) dε.

It is clear that P_{i} is a norm one projection, and P_{i}P_{j} = P_{j}P_{i} for all i, j = 1, …, m. Let Q_{i} = I − P_{i}. We have P_{i}ε_{i} = 0, P_{i}ε_{j} = ε_{j} for j ≠ i. For every α ⊂ {1, …, m} let P^{α} = ∏_{i∈α} P_{i}. We see by checking the action on every w_{α} that

T_{t} = ∏_{i=1}^{m}(P_{i} + e^{−t} Q_{i}) = ∏_{i=1}^{m}((1 − e^{−t})P_{i} + e^{−t} I).

It follows, by expanding the last product, that T_{t} is a convex combination of commuting norm one projections of the form P^{α}.

For the next lemma it is natural to quantify the type-p property of a Banach space X in a way close to the definition of B-convex Banach space. We let N(X) denote the smallest integer n ≥ 1 such that

∫_{0}^{1} ∥ ∑_{i=1}^{n} ε_{i}(t)x_{i}∥ dt ≤ n/16

for every family x_{1}, …, x_{n} of vectors in X such that ∥x_{i}∥ ≤ 1 for each i. Of course, if X has type p > 1, then we have N(X) ≤ (16 T_{p}(X))^{q}, where q < +∞ is the number conjugate to p > 1. We let N(X) = +∞ when X is not B-convex.

Lemma 9 Suppose that X is a B-convex Banach space, and assume that M is a convex combination of contractive commuting projections on X. Then

∥M^{n+1} − M^{n}∥ ≤ 1/4

when n ≥ max(N(X), 256).

PROOF. Let

M = ∑_{α} c_{α} P_{α}

where c_{α} ≥ 0, ∑_{α} c_{α} = 1, and where the (P_{α})s are commuting projections on X, such that ∥P_{α}∥ ≤ 1; we get in particular that ∥M∥ ≤ 1. Let ξ be a random variable on some probability space Ω, with values in the space of operators on X and with P (ξ = P_{α}) = c_{α} for every α. Then Eξ = M, and if ξ_{1}, ξ_{2} are two independent copies of ξ, then Eξ_{1}ξ_{2} = M^{2}. Let ξ_{1}, …, ξ_{n} be independent copies of ξ, with n ≥ max(N(X), 256). Suppose that x ∈ X, ∥x∥ = 1 and let us consider, for a fixed choice ε of ε_{i} = ±1, the random variable Z_{ε} on Ω defined by Z_{ε}(ω) = ∥∑_{i=1}^{n} ε_{i}ξ_{i}(ω)x∥.

Let B = {i : ε_{i} = 1} and C = {i : ε_{i} = −1}, k = |B| and ℓ = |C|. Assume that k ≤ ℓ. Then, letting ξ^{B} denote the (random) operator equal to ∏_{j∈B} ξ_{j}, and noting that ∥ξ^{B}(ω)∥ ≤ 1 for every ω,

∥∑_{i=1}^{n} ε_{i}ξ_{i}x∥ ≥ ∥ξ^{B}(∑_{i=1}^{n} ε_{i}ξ_{i}x)∥ = ∥∑_{i∈B} ξ^{B}x − ∑_{i∈C} ξ^{B}ξ_{i}x∥.

Taking expectation on Ω,

E ∥∑_{i=1}^{n} ε_{i}ξ_{i}x∥ ≥ ∥∑_{i∈B} E ξ^{B}x − ∑_{i∈C} E ξ^{B}ξ_{i}x∥
= ∥kM^{k}x − ℓM^{k+1}x∥ ≥ ℓ ∥M^{k}x − M^{k+1}x∥ − |ℓ − k|
≥ \frac{n}{2} ∥M^{n}x − M^{n+1}x∥ − |ℓ − k|.

(if ℓ ≤ k, we replace ξ^{B} by ξ^{C}). On the other hand we get, taking the expectation E′ over all signs, noting that |ℓ − k| = | ∑_{i=1}^{n} ε_{i}| and since n ≥ N(X)

\frac{n}{2} ∥M^{n}x − M^{n+1}x∥ − \sqrt{n} ≤ EE′ ∥ ∑_{i=1}^{n} ε_{i}ξ_{i}x∥ ≤ n/16.

so that, using n ≥ 256

∥M^{n}x − M^{n+1}x∥ ≤ 1/8 + 2n^{−1/2} ≤ 1/4.

Remark 10 Suppose that X is a type-p Banach space, with p > 1 and type-p constant T_{p}. Assume that M is a convex combination of contractive commuting projections on X. Then

∥x + M(x)∥ ≥ (4T_{p})^{−q}∥x∥

for every x ∈ X (q is the exponent conjugate to p). It follows that I + M is invertible and that ∥(I + M)^{−1}∥ ≤ (4T_{p})^{q}.

It is well known to experts that the uniform invertibility of I + T_{t} is precisely what is needed in Kato’s theorem for proving that a semi-group (T_{t})_{t≥0} is holomorphic. The proof of the remark is a slight modification of the proof of the preceding lemma. Suppose that ∥x∥ = 1 and ∥x + Mx∥ < ε. It follows that ∥M^{k}x + M^{k+1}x∥ < ε for every k ≥ 0 since ∥M∥ ≤ 1, and ∥M^{k}x∥ ≥ ∥x∥ − kε = 1 − kε by the triangle inequality. Taking expectations as before,

E ∥∑_{i=1}^{n} ε_{i}ξ_{i}x∥ ≥ ∥nM^{k}x∥ − ℓε ≥ n(1 − kε) − ℓε ≥ n − (n + 1)^{2}ε/2.

Taking the expectation E′ over all signs

n − n^{2}ε ≤ EE′ ∥ ∑_{i=1}^{n} ε_{i}ξ_{i}x∥ ≤ T_{p} n^{1/p}.

If we choose n such that 1/4 < nε ≤ 1/2, then n/2 ≤ T_{p} n^{1/p}, thus 4n ≤ (4T_{p})^{q} since q ≥ 2 and ε ≥ (4T_{p})^{−q}.

If we want to see why things can go wrong when X contains ℓ_{1}^{n}s, we may modify the example showing that the K-convexity constant of L_{1}(G^{m}) is large. We shall only sketch the idea. Let us consider the function f_{0} from [0, 1]^{m} to the space of measures on [0, 1]^{m} such that f_{0}(x) is the Dirac mass at x for every x ∈ [0, 1]^{m} (this function is not Bochner measurable; a genuine example should correct this fact). If P_{i} is defined for every g ∈ L_{2}([0, 1]^{m}, X) by

(P_{i}g)(x_{1}, …, x_{m}) = ∫_{0}^{1} g(x_{1}, …, x_{i−1}, y, x_{i+1}, …, x_{m}) dy

for i = 1, …, m, then the (P_{i}) are commuting norm one projections. For every α ⊂ {1, …, m}, the vector value (P^{α}f_{0})(x) is the Lebesgue measure on some |α|-dimensional unit cube. When x varies, these probability measures are pairwise disjoint, and this is the source of all the problems. The corresponding semi-group S_{t} = ∏_{i=1}^{m}((1 − e^{−t})P_{i} + e^{−t} I) behaves very badly. In particular, the inequality ∥I − S_{t}∥ ≥ 2 (1 − e^{−mt}) shows that the hypothesis for Kato’s theorem is not satisfied uniformly in m in this example, where X = M (the space of measures).

We are ready to begin the proof that B-convexity implies K-convexity, using the Walsh semi-group (T_{t})_{t≥0} defined by relation (W). Recall that each operator T_{t} is a convex combination of commuting norm one projections on L_{2}(X). If X is B-convex, then L_{2}(X) is also B-convex; it follows from Lemma 9 that

∥T_{nt} − T_{(n+1)t}∥ ≤ 1/4

for every t > 0, when n ≥ max(N(L_{2}(X)), 256). For the rest of the paper we assume that X is a complex Banach space.

We strongly recommend reading [MS, chapter 14] (and appendix IV about Kato’s theorem for semi-groups). For the lazy reader who does not want to hear about general semi-groups, we shall sketch a proof of Kato’s theorem in the simplified setting which is needed here. We consider m Rademacher functions ε_{1}, …, ε_{m} and the corresponding 2^{m} Walsh functions (w_{α}) that are defined by the formula w_{α} = ∏_{i∈α} ε_{i}, where α ranges over the 2^{m} subsets of {1, …, m}; next we fix 2^{m} vectors (y_{α}) in X, and we let E be the 2^{m}-dimensional complex subspace of L_{2}(G^{m}, X) generated by the algebraic basis (w_{α}y_{α}). Our operators (T_{t})_{t≥0} act diagonally on this basis of E, since T_{t}(w_{α}y_{α}) = e^{−t|α|} w_{α}y_{α} for every α. Defining the complex extension T_{z} of T_{t} on E is straightforward: we simply say that T_{z} acts on E by T_{z}(w_{α}y_{α}) = e^{−z|α|} w_{α}y_{α} for every α, but of course the problem is to find bounds for the norm of T_{z}, independent of the particular subspace E ⊂ L_{2}(X). We see that T_{t} = e^{−tA}, where A is represented in the basis (w_{α} y_{α}) by a diagonal matrix with entries in {0, 1, …, m}, namely A(w_{α} y_{α}) = |α| w_{α} y_{α}. The Rademacher projection corresponds to the matrix B obtained by replacing in A all diagonal entries ≠ 1 by zero entries.

For the proof of Theorem 12 below, we shall keep m and the 2^{m}-dimensional subspace E ⊂ L_{2}(G^{m}, X) fixed. Our aim is to find a bound for the norm of the matrix B, acting on this subspace E by B(w_{α}y_{α}) = w_{α}y_{α} if |α| = 1 and B(w_{α}y_{α}) = 0 otherwise; we are looking for a bound K independent of m and of the particular subspace E. From the nature of the problem it is clear that such a bound K will be a bound for the norm of the Rademacher projection R_{X} acting on L_{2}(G^{ℕ}, X), that is to say a bound for the K-convexity constant of X.

The control of the complex extension of the semi-group begins with a standard exercise in functions of one complex variable. Consider η = v + iπ, v > 0, and the two conjugate rays R = ℝ_{+}η and \overline{R} = ℝ_{+}\overline{η}, symmetric with respect to the real axis, contained in the half plane ℜz > 0. Let ξ = π + iu, with |u| < v, and consider the holomorphic function f(z) = e^{−ξz}. Then for every real a ≥ 0, we have

e^{−ξa} = \frac{1}{2iπ} ∫_{Γ} e^{−ξz}(z − a)^{−1} dz

where Γ is essentially the path given by these two rays, except for a little detour to avoid z = 0 (this is needed in the case a = 0; see the figure in [MS, appendix IV]). We have

| e^{−ξz} | ≤ e^{−π(1−|u|/v) ℜz}

for every z in the convex cone limited by R and \overline{R}, therefore the integral is convergent since |u| < v. It is a standard exercise to show that the integral over Γ is indeed equal to e^{−ξa} (approximate ∫_{Γ} by the integral over a bounded closed contour that uses part of the two rays and part of a large circle centered at 0, and apply Cauchy’s formula).

In our (finite-dimensional) vector situation, the generator A of the semi-group is expressed by a diagonal matrix with non-negative real diagonal, so that the next equation is by no means harder to prove than the scalar case,

e^{−ξA} = \frac{1}{2iπ} ∫_{Γ} e^{−ξz}(zI − A)^{−1} dz.

This can be done not only for ξ = π+iu, |u| < v, but as well for any ξ = α+iβ with α > 0 and π |β| < v α, in other words for every ξ in a sector of angle θ around the positive real axis, where π tan θ = v. The above formula, extended to these values of ξ, defines the complex extension of the semi-group. It is clear (and standard) that we can bound the complex extension of the semi-group, acting on the fixed finite-dimensional subspace E, if we have a suitable bound for the norm of the resolvent (zI − A)^{−1} on the two rays R and \overline{R} (again, this norm is understood as norm of an operator from E to E).

Lemma 11 Let E ⊂ L_{2}(G^{m}, X) be as above. Assume that X is a B-convex Banach space, let n ≥ max(N(L_{2}(X)), 256) and let v be such that 0 < v ≤ 1/n. For every complex number z belonging to the ray R = ℝ_{+}(v + iπ) or to the conjugate ray \overline{R}, we have

∥(zI − A)^{−1}∥ ≤ 36 π n/|ℑz|.

PROOF. Let λ = v ± iπ, and suppose that ε > 0 is chosen in such a way that ∥(A − λI)^{−1}∥ > 1/ε; we can find a norm one vector x ∈ E such that ∥Ax − λx∥ < ε. The function φ(t) = T_{t}(x) = e^{−tA} x satisfies the differential equation φ′(t) = −A φ(t) = −T_{t}(Ax). Since T_{t} is a contraction semi-group, we deduce that for every t > 0

∥φ′(t) + λφ(t)∥ = ∥T_{t}(Ax − λx)∥ ≤ ε.

If we write this as φ′(t)+λφ(t) = g(t) with ∥g(t)∥ ≤ ε and solve the differential equation, we get

φ(t) = e^{−λt}(x + ∫_{0}^{t} e^{λs} g(s) ds),

which implies that ∥φ(t) − e^{−λt} x∥ ≤ εt. Let n ≥ max(N(L_{2}(X)), 256). By Lemma 9, we know that for every s > 0, we have ∥T_{(n+1)s} − T_{ns}∥ ≤ 1/4, since T_{s} is a convex combination of commuting norm one projections. We shall use this fact with s = 1; when s = 1, we get ∥φ(n + 1) − φ(n)∥ ≤ 1/4 and e^{−λs} = e^{−λ} = − e^{−v} since e^{iπ} = −1. We have

1/3 < e^{−1} < e^{−vn}(1 + e^{−v}) = ∥ e^{−λ(n+1)} x − e^{−λn} x∥.

By the triangle inequality,

1/3 < ∥ e^{−λ(n+1)} x − e^{−λn} x∥
≤ ∥φ(n + 1) − e^{−λ(n+1)} x∥ + ∥φ(n) − e^{−λn} x∥ + 1/4,

hence

1/12 ≤ ∥φ(n + 1) − e^{−λ(n+1)} x∥ + ∥φ(n) − e^{−λn} x∥ ≤ (2n + 1)ε ≤ 3n ε.

It follows that ∥(A − λI)^{−1}∥ ≤ 36 n. We may apply the same proof to the generator A_{s} = s^{−1}A, for every s > 0; obviously, this A_{s} also generates a semigroup consisting of convex combinations of commuting contractive projections, and this implies as above that

∥(λI − A_{s})^{−1}∥ ≤ 36 n or ∥(sλI − A)^{−1}∥ ≤ 36 n/s

hence

∥(zI − A)^{−1}∥ ≤ 36 π n/|ℑz|

when z belongs to the rays R = ℝ_{+}(v + iπ) or \overline{R}.

Theorem 12 Let X be a B-convex Banach space. Then the Rademacher projection R_{X} is bounded on L_{2}(G^{ℕ}, X), and

∥R_{X}∥ ≤ e^{κ max(N(L_{2}(X)), 256)}

for some universal constant κ.

PROOF. Let us consider again the 2^{m}-dimensional subspace E of L_{2}(G^{m}, X). We may deduce from Lemma 11 that the semi-group (T_{t}) acting on E has a nicely bounded complex extension to the sector mentioned before, but since we are mainly interested in the Rademacher projection we are going to take a shortcut. Let N = max(N(L_{2}(X)), 256) and v = 1/N; consider the path Γ consisting of the ray R = ℝ_{+} (v + iπ) and its conjugate \overline{R}. Provided that the integral makes sense,

\frac{1}{2iπ} ∫_{Γ} φ(z) (zI − A)^{−1} dz

represents, when φ is holomorphic on ℂ, the diagonal matrix where each diagonal entry k of A is replaced by φ(k). Recall that the diagonal entries of A are integers. In order to get the matrix B of the Rademacher projection, we naturally introduce φ_{1}(z) = sin(πz)/(1 − z) that kills all entries ≠ 1 in A. We need to multiply this φ_{1} by a suitable exponential that guarantees that the integral converges and that the Cauchy formula applies to the unbounded contour Γ. Let us consider the matrix

C = \frac{1}{2iπ} ∫_{Γ} \frac{sin(πz)}{π(1 − z)} e^{−π^{2}z/v} (zI − A)^{−1} dz.

The sin function eliminates the problem at 0. Also, one can check that the integral is absolutely convergent. It is easy to see that this matrix C is a multiple of the Rademacher projection B, namely C = e^{−π^{2}/v} B and using the bound from Lemma 11 we can show that

∥C∥ ≤ κ_{1}N,

where κ_{1} is an universal constant. It follows that ∥B∥ ≤ κ_{1}N e^{π^{2}N} ≤ e^{κ N}.

Let us detail the preceding computation. We have N = max(N(L_{2}(X)), 256) and v = 1/N, thus 0 < v < 1/2. Let z_{0} = v ± iπ. If z = s z_{0}, s > 0, then | sin(πz)| ≤ e^{π^{2}s} and | cos(πz)| ≤ e^{π^{2}s}, therefore | sin(πz) e^{−π^{2}z/v} | ≤ 1. We also have |1 − z| ≥ πs and |1 − z| ≥ 1 − sv ≥ 1/2 when 0 < s < 1. Next, we use

| sin(πz)| ≤ π|z| max_{0<u<1} | cos(uπz)| ≤ πs |z_{0}| e^{π^{2}s}

for 0 < s < 1, so that

∥C∥ ≤ \frac{1}{π} ∫_{0}^{1} \frac{2πs|z_{0}|}{π} \frac{36N}{s} |z_{0}| ds + \frac{1}{π} ∫_{1}^{∞} \frac{1}{π^{2}s} \frac{36 N}{s} |z_{0}| ds

≤ \frac{72 N |z_{0}|^{2}}{π} + \frac{36 N |z_{0}|}{π^{3}} ≤ 500 N.

We finish with another result of Pisier, that has been very useful for local theory. We did not try to optimize the constant, but to give an argument as simple as possible (essentially identical to Pisier’s proof).

Theorem 13 Let d_{X} denote the Banach-Mazur distance from X to the Hilbert space of the same dimension. Then the norm of the Rademacher projection in L_{2}(X) is bounded by 4 ln d_{X} when d_{X} ≥ e (and by d_{X} in any case).

PROOF. The proof is comparatively simple. Let f = ∑_{α} w_{α}x_{α} be a function from G^{m} to X. Assume that ∥f∥_{L_{2}(X)} = 1. For z ∈ ℂ, let

P (z) = ∑_{α} z^{|α|}w_{α}x_{α}.

This defines a holomorphic function (a polynomial) from ℂ to L_{2}(X). The Rademacher projection R_{X}(f) of f is the derivative P′(0) of P at z = 0. When X = H is a Hilbert space, we get by orthogonality, for every z in the closed unit disc D in ℂ

∥P (z)∥_{L_{2}(H)}^{2} = ∑_{α} |z|^{2|α|}∥x_{α}∥^{2} ≤ ∑_{α} ∥x_{α}∥^{2} = ∥f∥_{L_{2}(H)}^{2},

thus ∥P (z)∥ ≤ 1 for every z ∈ D in this case. If the distance from X to some Hilbert space is ≤ d, then clearly ∥P (z)∥ ≤ d, therefore ∥P (z)∥ ≤ d_{X} for every z ∈ D. On the other hand, we have seen that ∥P (x)∥_{L_{2}(X)} ≤ ∥f∥_{L_{2}(X)} = 1 when x is real and |x| ≤ 1, because P (x) = μ_{x} ∗ f in that case, with μ_{x} a probability measure. For convenience, we transfer the problem to the closed strip S = {z : |ℑz| ≤ 1}: the mapping φ(z) = tanh(πz/4) maps S to the closed unit disc, and sends the line ℑz = 0 to the segment −1 ≤ x ≤ 1. The L_{2}(X)-valued function q(z) = P (φ(z)) is bounded on S, holomorphic on the open strip S_{0}, bounded by d_{X} on S and by 1 on the line ℑz = 0. The result follows then from q′(0) = πP′(0)/4 and from the following lemma. When d_{X} ≥ e we get |P′(0)| = 4 π^{−1} |g′(0)| ≤ 4 π^{−1} e ln d_{X} < 4 ln d_{X}.

Lemma 14 Let g be a bounded and continuous function, defined on the closed strip S = {z : |ℑz| ≤ 1}, holomorphic on the open strip S_{0}, with values in a Banach space Y; assume that g is bounded by C ≥ e on S and bounded by 1 on the line ℑz = 0. Then |g′(0)| ≤ e ln C.

PROOF. Let g_{1}(z) = (g(z) − g(−z))/2; then g_{1} obeys the same bounds as does g, and g′_{1}(0) = g′(0); furthermore, g_{1}(0) = 0. Let 0 < θ ≤ 1. Since g_{1} is bounded, |g_{1}| ≤ 1 on the line ℑz = 0 and |g_{1}| ≤ C on the line ℑz = 1, the three lines Lemma implies that |g_{1}| is bounded by 1^{1−θ}C^{θ} = C^{θ} on the line ℑz = θ; the same argument applies to the line ℑz = −θ; now k(z) = g_{1}(z)/z (with k(0) = g′_{1}(0)) is bounded on the strip S, and bounded by θ^{−1}C^{θ} on the two lines ℑz = ±θ, therefore |g′(0)| = |g′_{1}(0)| = |k(0)| ≤ θ^{−1}C^{θ}. The optimal choice of θ in (0, 1] is θ = (ln C)^{−1}, which is licit because ln C ≥ 1.

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