Martingale Inequalities and

Operator Space Structures on L_{p}

Gilles Pisier^{1}

Received: September 10, 2012

Communicated by Joachim Cuntz

Abstract. We describe a new operator space structure on L_{p} when p is an even integer and compare it with the one introduced in our previous work using complex interpolation. For the new structure, the Khintchine inequalities and Burkholder’s martingale inequalities have a very natural form: the span of the Rademacher functions is completely isomorphic to the operator Hilbert space OH, and the square function of a martingale difference sequence d_{n} is Σ d_{n} ⊗ d̄_{n}. Various inequalities from harmonic analysis are also considered in the same operator valued framework. Moreover, the new operator space structure also makes sense for non-commutative L_{p}-spaces associated to a trace with analogous results. When p → ∞ and the trace is normalized, this gives us a tool to study the correspondence E ↦ \underline{E} defined as follows: if E ⊂ B(H) is a completely isometric emdedding then \underline{E} is defined so that \underline{E} ⊂ CB(OH) is also one.

2010 Mathematics Subject Classification: Primary 47L07,46L53; Secondary 46B28,60G48,47L25.

Contents

1 Background on operator spaces  1372
2 Preliminary results  1375
3 Definition of Λ_{2m} Λ_{2m} Λ_{2m}  1379
4 Martingale inequalities in Λ_{p} Λ_{p} Λ_{p}  1385

1Partially supported by ANR-2011-BS01-008-01.

Documenta Mathematica 19 (2014) 1367–1442

5 Burkholder-Rosenthal inequality  1395
6 Hilbert transform  1399
7 Comparison with L_{p} L_{p} L_{p}  1401
8 The non-commutative case  1404
9 Comparisons  1409
10 Connection with CB maps on OH  1412
11 Non-commutative Khintchine inequalities  1418
12 Non-commutative martingale inequalities  1425
13 p-orthogonal sums  1428
14 Lacunary Fourier series in Λ_{p}  1432
15 Appendix  1436

Introduction

In probability theory and harmonic analysis, the classical inequalities for martingales in L_{p} due to Don Burkholder, and also to Richard Gundy and Burgess Davis have had an invaluable impact, with multiple interaction with other fields. See [8] for a recent review. In [26], a non-commutative version of Burkholder’s martingale inequalities is given. This is valid in any non-commutative L_{p}-space say L_{p}(τ) (associated to a finite trace τ on a von Neumann algebra) for any 1 < p < ∞. In particular this applies to martingales of the form f_{n} = ∑_{1}^{n} ε_{k} ⊗ x_{k} where x_{k} ∈ L_{p}(τ) and (ε_{n}) is a standard random choice of signs ε_{n} = ±1 (equivalently we can think of (ε_{n}) as the Rademacher functions on [0,1]). In that case, Burkholder’s inequality reduces to Khintchine’s inequality, for which the non-commutative case is due to Lust–Piquard ([20]). In the classical setting, Khintchine’s inequality expresses the fact that the closed span in L_{p} of {ε_{n}} is isomorphic to ℓ_{2} (as a Banach space). If {ε_{n}} is replaced by a sequence (g_{n}) of independent standard Gaussian random variables, the span in L_{p} becomes isometric to ℓ_{2}. In the recently developed theory of operator spaces, Lust–Piquard’s non-commutative Khintchine inequalities can be interpreted (see [23, p. 108]) as saying that the span in L_{p} of [ε_{n}] (or (g_{n})) is completely isomorphic to a Hilbertian operator space that we will denote here by KH_{p}. The precise description of KH_{p} is not important for this paper, but for reference let us say merely that, for 2 < p < ∞ (resp. 1 ≤ p < 2), KH_{p} has the structure of intersection (resp. sum) of row and column spaces in the Schatten class S_{p}. For the non-commutative Burkholder inequalities from [26], the situation is analogous: the relevant square function combines the two cases of “row” and “column” in analogy with the definition of KH_{p}. This result was a bit of a disappointment because there is a canonical notion of “operator Hilbert space,” namely the space OH from [22] and one would have expected in closer analogy to the classical case, that the span in L_{p} of [ε_{n}] (or (g_{n})) should be completely isomorphic to OH. In the preceding references, the spaces L_{p} (commutative or not) are always equipped with what we call their “natural” operator space structure defined using complex interpolation. Then the space KH_{p} is completely isomorphic to OH only when p = 2. In the present paper, we take a different route. We will equip L_{p} with another o.s.s., hopefully still rather natural, but limited to p equal to an even integer (for some of our results we even assume p = 2^{k}). Roughly we imitate the classical idea that f ∈ L_{4} iff |f|^{2} ∈ L_{2} in order to define our new o.s.s. on L_{4} and then we iterate the process to define the same for L_{6}, L_{8} and so on. Thus given the space L_{p}(Ω, μ) we can associate to it (assuming p ∈ 2ℕ and p ≥ 2) an operator space that we denote by Λ_{p}(Ω, μ) that is isometric to the original L_{p}(Ω, μ) as a Banach space. It turns out that with this new structure a quite different picture emerges for Khintchine’s (or more generally Burkholder’s) inequalities. Indeed, we will prove that the span of {ε_{n}} in Λ_{p}(Ω, μ) is completely isomorphic to the space OH, (i.e. to ℓ_{2} equipped with the o.s.s. of OH). Similarly we will prove martingale inequalities involving a square function that is simply defined as S = ∑ d_{n} ⊗ d̄_{n} when (d_{n}) is a martingale difference sequence. We limit our treatment to L_{p} for p an even integer. Thus we stopped shy of making the obvious extensions: we can use duality to define Λ_{q} for 1 < q < 2 of the form q = \frac{2n}{2n−1} for some integer n > 1 and then use complex interpolation to define Λ_{p} in the remaining intermediate values of p’s or q’s. While this procedure makes perfectly good sense it is rather “unnatural” given that if p(0), p(1) and p(θ) are even integers such that p_{0} < p_{θ} < p_{1} and p(θ)^{−1} = (1 − θ)p(0)^{−1} + θp(1)^{−1}, the space Λ_{p(θ)} does not coincide (in general) with (Λ_{p(0)}, Λ_{p(1)})_{θ}. This happens for instance when p(0) = 2, p(1) = ∞ and θ = 1/2, since the spaces L_{4} and Λ_{4} differ as operator spaces. We will now review the contents of this paper. After some general background in §1, we explain in §2 some basic facts that will be used throughout the paper. The main point is that we use an ordering on B(H) ⊗ \overline{B(H)} denoted by T ≺ S that is such that

0 ≺ T ≺ S ⇒ ∥T ∥ ≤ ∥S∥

where the norm is the minimal (or spatial) tensor product on B(H)⊗_{min} \overline{B(H)}, i.e. the norm induced by B(H ⊗_{2} \overline{H}). As we explain in §2, it is convenient to abuse the notation and to extend the notation T ≺ S to pairs, T, S in B(H_{1}) ⊗ ⋯ ⊗ B(H_{2n}) when the collection {H_{1}, …, H_{2n}} can be permuted to be of the form \{K_{1}, …, K_{n}, \overline{K}_{1}, …, \overline{K}_{n}\} so that T, S can be identified with elements of B(H) ⊗ \overline{B(H)} with H = K_{1} ⊗ ⋯ ⊗ K_{n}.

In §3, we use the properties of this ordering to prove a version of Hölder’s inequality that allows us to introduce, for each even integer p, a new operator space structure on the space L_{p}(Ω, A, μ) associated to a general measure space (Ω, A, μ). We follow the same route that we used for p = 2 in [22] to define the space OH starting from Haagerup’s Cauchy–Schwarz inequality. Suitable iterations of the latter leads to versions of Hölder’s inequality in L_{4}, L_{6}, L_{8}, … from which a specific norm can be introduced on B(H) ⊗ L_{p}(μ) that endows L_{p}(μ) with an operator space structure. We denote by Λ_{p}(Ω, Σ, μ) the resulting operator space. It is natural to identify Λ_{1}(Ω, Σ, μ) with L_{1}(μ) equipped with its maximal operator space structure in the Blecher-Paulsen sense (see e.g. [11, 25] for details on this). Some of the calculations involving Λ_{p}(μ) are rather satisfactory, e.g. for any f_{j} ∈ B(H_{j}) ⊗ L_{p}(μ), j = 1, …, p the pointwise product L_{p} × ⋯ × L_{p} → L_{1} applied to (f_{1}, …, f_{n}) leads to an element denoted by F = f_{1} ˙⊗ ⋯ ˙⊗f_{n} in B(H_{1}) ⊗ ⋯ ⊗ B(H_{n}) ⊗ L_{1}, and if H = H_{1} ⊗ ⋯ ⊗ H_{n} we have

∥F ∥_{B(H)⊗_{min}Λ_{1}(μ)} ≤ Π∥f_{j}∥_{B(H_{j})⊗_{min}Λ_{p}}.

In particular, if q ≤ p are even integers and if μ(Ω) = 1, the inclusion Λ_{p}(μ) → Λ_{q}(μ) is completely contractive. We also show that all conditional expectations are completely contractive on Λ_{p}(μ). When p → ∞, we recover the usual operator space structure of L_{∞}(μ) as the limit of those of Λ_{p}(μ). In §4 we prove a version of Burkholder’s square function inequalities for martingales in Λ_{p}(μ). If (d_{n}) is a sequence of martingale differences in B(H) ⊗ L_{p} the relevant square function is S = Σd_{n} ˙⊗d̄_{n}. Here we restrict to p = 2^{k} for some k, but at least one side of the inequality is established for any even p by a different argument in §13. We also prove an analogue for Λ_{p} of the inequality due to Stein expressing that for any sequence (f_{n}) in L_{p} the L_{p}-norm of (∑ |f_{n}|^{2})^{1/2} dominates that of (∑ |𝔼_{n}(f_{n})|^{2})^{1/2} for any 1 < p < ∞. In §5, we consider the conditioned square function σ = ∑ 𝔼_{n−1}(d_{n} ˙⊗d̄_{n}) and we prove a version of the Burkholder–Rosenthal inequality adapted to Λ_{p}(μ). As can be expected, the preceding inequalities imply the complete boundedness of the multipliers called “martingale transforms”. Not surprisingly, in §6 we can also prove similar results for the Hilbert transform, say on 𝕋 or ℝ, using the well known Riesz-Cotlar trick. In §7, we compare the “old” and the “new” o.s.s. on L_{p}(μ). We show that the (isometric) inclusion L_{p}(μ) → Λ_{p}(μ) is completely contractive, but its inverse is not completely bounded and we show that its c.b. norm in the n-dimensional case grows at least like n^{\frac{1}{2} ( \frac{1}{2} − \frac{1}{p})}. In §8, we turn to the non-commutative case. We introduce the space Λ_{p}(τ) associated to a non-commutative measure space (M, τ). By this we mean a von Neumann algebra M equipped with a semi-finite faithful normal trace τ. In §9, we repeat the comparison made in §7. It turns out that the noncommutative case is significantly more intricate, mainly because the (joint) complete boundedness of the product map L_{p} × L_{q} → L_{r} (p^{−1} + q^{−1} = r^{−1}) no longer holds in general (I am grateful to Quanhua Xu for drawing my attention to this). This leads us to consider yet another operator space structure on L_{p}(τ) that we denote by L_{p}(τ) for which it still holds (see Proposition 9.1). When p ∈ 2ℕ, we then extend the main result of §7 by showing (see Corollary 9.2) that the identity defines a completely contractive map L_{p}(τ) ∩ L_{p}(τ)^{op} → Λ_{p}(τ). In the commutative case L_{p}(τ) and L_{p}(τ) are identical. We give estimates of the growth in n of the c.b. norms of the maps L_{p}(M_{n}, tr) → Λ_{p}(M_{n}, tr) and Λ_{p}(M_{n}, tr) → L_{p}(M_{n}, tr) induced by the identity. In §10, assuming τ(1) = 1, we study the “limit” when p → ∞ of the spaces Λ_{p}(M, τ), that we denote by Λ_{∞}(M, τ). Surprisingly, we are able to identify the resulting operator space: Indeed, when (M, τ) is M_{n} equipped with its normalized trace then Λ_{∞}(M, τ) can be identified completely isometrically with CB(OH_{n}, OH_{n}), i.e. the space of c.b. maps on the n-dimensional operator Hilbert space. More generally (see Theorem 10.3), when M ⊂ B(H), the o.s.s. of Λ_{∞}(M, τ) can be identified with the one induced by CB(OH, OH), where by OH we mean H equipped with its unique self-dual structure in the sense of [25]. The verification of these facts leads us to several observations on the space CB(OH, OH) that may be of independent interest. In particular, the latter space satisfies a curious identity (see (10.4)) that appears like an operator space analogue of the Gelfand axiom for C^{∗}-algebras. Furthermore, to any operator space E ⊂ B(H), we associate the operator space \underline{E} ⊂ CB(OH) (equipped with the operator space structure induced by CB(OH)), and we show that if F is another operator space, for any cb-map u : E → F we have

∥u : \underline{E} → \underline{F} ∥_{cb} ≤ ∥u : E → F ∥_{cb}.

In §12 we extend the Burkholder inequalities except that—for the moment— we can only prove the two sides of the martingale inequality for p = 4. Note however that the right hand side is established for all even p in §13 by a different method based on the notion of p-orthogonality. Nevertheless, in §11, using Buchholz’s ideas in [7] we can prove versions of the non-commutative Khintchine inequalities for Λ_{p}(τ) with optimal constants for any even integer p. We may consider spin systems, free semi-circular (or “free-Gaussian”) families, or the free generators of the free group in the associated free group factor. Returning to the commutative case this yields the Rademacher function case with optimal constants. The outcome is that the span of each of these sequences in Λ_{p} is completely isomorphic to OH and completely complemented. In §14 we transplant the results of [13] (see also [14]) on non-commutative lacunary series to the setting of Λ_{p}(τ). We use the view point of [24] to abbreviate the presentation. Let Γ be a discrete group. We will consider Λ(p)-sets in Rudin’s sense inside Γ. The main point is that a certain class of Λ(p)-subsets of Γ again spans a copy of OH in the operator spaces Λ_{p}(M, τ) when M is the von Neumann algebra of Γ. For our new o.s.s. the relevant notion of Λ(p)-set is slightly more general than the one needed in [13].

Lastly in the appendix §15 we include a discussion of elements that have p-th moments defined by pairings as in Buchholz’s paper [7]. We simply translate in the abstract language of tensor products some very well known classical ideas on Wick products for Gaussian random variables. Our goal is to emphasize the similarity between the Gaussian case and the free or q-Gaussian analogues. We feel this appendix fits well with the extensive use of tensor products throughout the sections preceding it. The proof of the initial non-commutative martingale inequalities of [26] is the main source of inspiration for the present results. We also make crucial use of our version for the space Λ_{p} of Junge’s “dual Doob” inequality from [15]. Although we take a divergent route, we should point out to the reader that the methods of [26] have been considerably improved in a series of important later works, such as [17, 18, 19], by M. Junge and Q. Xu, or [28, 29, 21] by N. Randrianantoanina and J. Parcet. See also [16, 30, 31, 32, 37] for progress related to Khintchine’s inequalities. The reader is referred to these papers to get an idea of what the “main stream” on non-commutative martingale and Khintchine inequalities is about.

1 Background on operator spaces

In this section we summarize the Theory of Operator Spaces. We refer either to [11] or [25] for full details. We recall that an operator space is just a closed subspace of the algebra B(H) of all bounded operators on a Hilbert space H. Given an operator space E ⊂ H, we denote by M_{n}(E) the space of n × n matrices with entries in E and we equip it with the norm induced by that of M_{n}(B(H)), i.e. by the operator norm on H ⊕ ⋯ ⊕ H (n times). We denote by E ⊗ F the algebraic tensor product of two vector spaces E, F. If E ⊂ B(H) and F ⊂ B(K) are operator spaces, we denote by E ⊗_{min} F the closure of E ⊗F viewed as a subspace of B(H ⊗_{2} K). We denote by ∥ ∥_{min} the norm induced by B(H ⊗_{2} K) on E ⊗F or on its closure E ⊗_{min} F. A linear map u: E → F between operator spaces is called completely bounded (c.b. in short) if the associated maps u_{n}: M_{n}(E) → M_{n}(F) defined by u_{n}([x_{ij}]) = [u(x_{ij})] are bounded uniformly over n, and we define

∥u∥_{cb} = sup_{n≥1} ∥u_{n}∥.

We say that u is completely isometric if u_{n} is isometric for any n ≥ 1 and that u is a complete isomorphism if it is an isomorphism with c.b. inverse. By a well known theorem due to Ruan (see [11, 25]), an operator space E can be characterized up to complete isometry by the sequence of normed spaces {M_{n}(E) | n ≥ 1}. The data of the sequence of norms on the spaces M_{n}(E) (n ≥ 1) constitutes the operator space structure (o.s.s. in short) on the vector space underlying E. Note that M_{n}(E) = B(H_{n})⊗_{min} E where H_{n} denotes here the n-dimensional Hilbert space.

Actually, the knowledge of the o.s.s. on E determines that of the norm on B(H) ⊗ E for any Hilbert space H. Therefore we (may and) will take the viewpoint that the o.s.s. on E consists of the family of normed spaces (before completion)

(B(H) ⊗ E, ∥ · ∥_{min}),

where H is an arbitrary Hilbert space. The reader should keep in mind that we may restrict either to H = ℓ_{2} or to H = ℓ_{2}^{n} with n ≥ 1 allowed to vary arbitrarily. (If we fix n = 1 everywhere, the theory reduces to the ordinary Banach space theory.) To illustrate our viewpoint we note that

∥u∥_{cb} = sup_{H} ∥u_{H}: B(H) ⊗_{min} E → B(H) ⊗_{min} F ∥

where the mapping u_{H} is the extension (by density) of id ⊗ u. The most important examples for this paper are the spaces L_{p}(μ) associated to a measure space (Ω, A, μ). Our starting point will be the 3 cases p = ∞, p = 1 and p = 2. For p = ∞, the relevant norm on B(H) ⊗ L_{∞}(μ) is the unique C^{∗}-norm, easily described as follows: any f in B(H) ⊗ L_{∞}(μ) determines a function f: Ω → B(H) taking values in a finite dimensional subspace of B(H) and we have

∥f∥_{B(H)⊗_{min}L_{∞}(μ)} = ess sup_{ω∈Ω}∥f(ω)∥_{B(H)}.  (1.1)

For p = 1, the relevant norm on B(H) ⊗ L_{1}(μ) is defined using operator space duality, but it can be explicitly written as follows

∥f∥_{B(H)⊗_{min}L_{1}(μ)} = sup ∥∫ f(t) ⊗ g(t) dμ(t)∥_{B(H⊗_{2}K)}  (1.2)

where the sup runs over all g in the unit ball of (B(K) ⊗ L_{∞}(μ), ∥ · ∥_{min}) and over all possible K. Equivalently, we may restrict to H = K = ℓ_{2}. This is consistent with the standard dual structure on the dual E^{∗} of an operator space. There is an embedding E^{∗} ⊂ B(H) such that the natural identification

B(H) ⊗ E^{∗} ⟷ B(E, B(H))

defines for any H an isometric embedding

B(H) ⊗_{min} E^{∗} ⊂ CB(E, B(H)).

With this notion of duality we have L_{1}(μ)^{∗} = L_{∞}(μ) completely isometrically. Moreover the inclusion L_{1}(μ) ⊂ L_{∞}(μ)^{∗} is completely isometric, and this is precisely reflected by the formula (1.2). To define the o.s.s. on L_{2}(μ), we will use the complex conjugate \overline{H} of a Hilbert space H. Note that the map x → x^{∗} defines an anti-isomorphism on B(H). Since \overline{B(H)} = B(\overline{H}) (canonically), we may view x → x^{∗} as a linear ∗-isomorphism from \overline{B(H)} = B(\overline{H}) to B(H)^{op} where B(H)^{op} is the same C^{∗}-algebra as B(H) but with reversed product. Then for any x_{k} ∈ B(H), y_{k} ∈ B(K) we have

∥∑ x_{k} ⊗ y_{k}∥_{min} = sup {∥(∑ x_{k} ⊗ y_{k}) (ξ)∥_{H⊗_{2}K} | ξ ∈ H ⊗ K, ∥ξ∥_{H⊗_{2}K} ≤ 1}.

Let us denote by S_{2}(H, K) the class of Hilbert–Schmidt operators from H to K with norm denoted by ∥ · ∥_{S_{2}(H,K)} or more simply by ∥ ∥_{2}. We may identify canonically ξ ∈ H ⊗_{2} K with an element ξ̂: K^{∗} → H with Hilbert–Schmidt norm ∥ξ̂∥_{2} = ∥ξ∥_{H⊗_{2}K}. Then for any y ∈ B(K) let ^{t}y: K^{∗} → K^{∗} denote the adjoint operator. We have then

∥∑ x_{k} ⊗ y_{k}∥_{min} = sup {∥∑ x_{k}ξ̂^{t}y_{k}∥_{2} | ∥ξ̂∥_{2} ≤ 1} .

Using the identification \overline{B(K)} = B(K)^{op} via x̄ → x^{∗}, (let x → x̄ be the identity map on B(K) viewed as a map from B(K) to \overline{B(K)}) we find

∥∑ x_{k} ⊗ ȳ_{k}∥_{B(H⊗_{2}\overline{K})} = sup \{∥∑ x_{k}ay_{k}^{∗}∥_{2} | a ∈ S_{2}(K, H), ∥a∥_{2} ≤ 1\} .

We now define the “natural” o.s.s. on the space ℓ_{2} according to [22]. This is defined by the following formula: for any f in B(H) ⊗ ℓ_{2}, of the form f = ∑_{1}^{n} x_{k} ⊗ e_{k} (here (e_{k}) denotes the canonical basis of ℓ_{2}) we have

∥f∥_{B(H)⊗_{min}ℓ_{2}} = ∥∑ x_{k} ⊗ x̄_{k}∥_{B(H⊗_{2}\overline{H})}^{1/2} .  (1.3)

The resulting o.s. is called “the operator Hilbert space” and is denoted by OH. Actually, the same formula works just as well for any Hilbert space H with an orthonormal basis (e_{i})_{i∈I}. The resulting o.s. will be denoted by H_{oh} (so that OH is just another notation for (ℓ_{2})_{oh}). In this paper our main interest will be the space L_{2}(μ). The relevant o.s.s. can then be described as follows: for any f in B(H) ⊗ L_{2}(μ) we have

∥f∥_{B(H)⊗_{min}L_{2}(μ)_{oh}} = ∥∫ f(ω) ⊗ \overline{f(ω)}dμ(ω)∥_{B(H⊗_{2}\overline{H})}^{1/2} .  (1.4)

It is not hard to see that this coincides with the definition (1.3) when (e_{n}) is an orthonormal basis of L_{2}(μ). We refer the reader to [22] for more information on the space H_{oh}, in particular for the proof that this space is uniquely characterized by its self-duality in analogy with Hilbert spaces among Banach spaces. We note that H_{oh} and H_{oh} are completely isometric iff the Hilbert spaces H and H are isometric (i.e. of the same Hilbertian dimension). The “natural o.s.s.” on L_{p} = L_{p}(μ) is defined in [23] for 1 < p < ∞ using complex interpolation. It is characterized by the following isometric identity: For any finite dimensional Hilbert space H

B(H) ⊗_{min} L_{p} = (B(H) ⊗_{min} L_{∞}, B(H) ⊗_{min} L_{1})_{1/p}.

When p = 2 we recover the o.s.s. defined above for L_{2}(μ)_{oh}. We now turn to multilinear maps. Let E_{1}, …, E_{m} and F be operator spaces. Consider an m-linear map φ: E_{1} × ⋯ × E_{m} → F. Let H_{1}, …, H_{m} be Hilbert spaces. We set B_{j} = B(H_{j}). By multilinear algebra we can associate to φ an m-linear map

φ̂: B_{1} ⊗ E_{1} × ⋯ × B_{m} ⊗ E_{m} → B_{1} ⊗ ⋯ ⊗ B_{m} ⊗ F

characterized by the property that (∀b_{j} ∈ B_{j}, ∀e_{j} ∈ E_{j})

φ̂(b_{1} ⊗ e_{1}, ⋯, b_{m} ⊗ e_{m}) = b_{1} ⊗ ⋯ ⊗ b_{m} ⊗ φ(e_{1}, ⋯, e_{m}).

We say that φ is (jointly) completely bounded (c.b. in short) if φ̂ is bounded from

B_{1} ⊗_{min} E_{1} × ⋯ × B_{m} ⊗_{min} E_{m} to B_{1} ⊗_{min} ⋯ ⊗_{min} B_{m} ⊗_{min} F.

It is easy to see that we may reduce to the case when H_{1} = H_{2} = ⋯ = H_{m} = ℓ_{2} (equivalently we could restrict to finite dimensional Hilbert spaces of arbitrary dimension). With this choice of H_{j} we set

∥φ∥_{cb} = ∥φ̂∥.

2 Preliminary results

We first recall Haagerup’s version of the Cauchy–Schwarz inequality on which is based a lot of what follows. Let H, K be Hilbert spaces, a_{k} ∈ B(H), b_{k} ∈ B(K) (k = 1, …, n). We have then

∥∑ a_{k} ⊗ b̄_{k}∥_{B(H⊗_{2}\overline{K})} ≤ ∥∑ a_{k} ⊗ ā_{k}∥_{B(H⊗_{2}\overline{H})}^{1/2} ∥∑ b_{k} ⊗ b̄_{k}∥_{B(K⊗_{2}\overline{K})}^{1/2}

where ⊗_{2} denotes the Hilbert space tensor product. We will sometimes need the following reformulation: let H be another Hilbert space and for any f ∈ H ⊗ B(H), say f = ∑ x_{k} ⊗ a_{k}, and g ∈ H ⊗ B(K), say g = ∑ y_{ℓ} ⊗ b_{ℓ} let ⟨⟨f, g⟩⟩ ∈ B(H) ⊗ \overline{B(K)} be defined by

⟨⟨f, g⟩⟩ = ∑_{k,ℓ}⟨x_{k}, y_{ℓ}⟩a_{k} ⊗ b̄_{ℓ}.

We have then

∥⟨⟨f, g⟩⟩∥_{B(H⊗_{2}\overline{K})} ≤ ∥⟨⟨f, f⟩⟩∥^{1/2}∥⟨⟨g, g⟩⟩∥^{1/2}.  (2.1)

More generally for any finite sequences (f_{α})_{1≤α≤N} in H⊗B(H) and (g_{α})_{1≤α≤N} in H ⊗ B(K) we have

∥∑_{α}⟨⟨f_{α}, g_{α}⟩⟩∥_{B(H⊗_{2}\overline{K})} ≤ ∥∑_{α}⟨⟨f_{α}, f_{α}⟩⟩∥_{B(H⊗_{2}\overline{H})}^{1/2} ∥∑_{α}⟨⟨g_{α}, g_{α}⟩⟩∥_{B(K⊗_{2}\overline{K})}^{1/2} .

(2.2)

It is convenient to also observe here that if E_{1}, E_{2} are orthogonal subspaces of H and if f_{j} ∈ E_{j} ⊗ B(H) (j = 1, 2) then

⟨⟨f_{1} + f_{2}, f_{1} + f_{2}⟩⟩ = ⟨⟨f_{1}, f_{1}⟩⟩ + ⟨⟨f_{2}, f_{2}⟩⟩.  (2.3)

We will use an order on B(H) ⊗ \overline{B(H)}: Let C_{+} be the set of all finite sums of the form ∑ a_{k} ⊗ ā_{k}. If x, y are in B(H) ⊗ \overline{B(H)}, we write x ≺ y (or y ≻ x) if y − x ∈ C_{+}. In particular x ≻ 0 means x ∈ C_{+}. Any element x ∈ B(H)⊗\overline{B(H)} defines a (finite rank) sequilinear form x̃: B(H)^{∗} ×B(H)^{∗} → ℂ. More generally, for any complex vector space E, we may define similarly x ≻ 0 for any x ∈ E ⊗ Ē. The following criterion is easy to show by linear algebra.

Lemma 2.1. Let x ∈ B(H) ⊗ \overline{B(H)}. Then x ∈ C_{+} iff x̃ is positive definite i.e. x̃(ξ, ξ) ≥ 0 for any ξ in B(H)^{∗}. Moreover, this holds iff x̃(ξ, ξ) ≥ 0 for any ξ in the predual B(H)_{∗} ⊂ B(H)^{∗} of B(H). Lastly, if H is separable, there is a countable subset D ⊂ B(H)_{∗} such that x ≻ 0 iff x̃(ξ, ξ) ≥ 0 for any ξ in D.

Proof. The first part is a general fact valid for any complex Banach space E in place of B(H): Assume x ∈ E ⊗ Ē, then x ∈ F ⊗ F̄ for some finite dimensional F ⊂ E, thus the equivalence in Lemma 2.1 just reduces to the classical spectral decomposition of a positive definite matrix. If E is a dual space with a predual E_{∗} ⊂ E^{∗}, then E_{∗} is σ(E^{∗}, E)-dense in E^{∗}, so the condition x̃(ξ, ξ) ≥ 0 will hold for any ξ ∈ E^{∗} if it does for any ξ ∈ E_{∗}. When the predual E_{∗} is separable, the last assertion becomes immediate.

Remark 2.2. Let E be a complex Banach space. Let [a_{ij}] be a complex n × n matrix, x_{j} ∈ E. Consider x = ∑ a_{ij}x_{i} ⊗ x̄_{j} ∈ E ⊗ Ē. Then x ≻ 0 if [a_{ij}] is positive definite, and if the x_{j}’s are linearly independent, the converse also holds. Indeed, by the preceding argument, all we need to check is x̃(ξ, ξ) = ∑ a_{ij}ξ(x_{i})\overline{ξ(x_{j})} ≥ 0.

The importance of this ordering for us lies in the following fact:

Lemma 2.3. If x, y ∈ B(H) ⊗ \overline{B(H)} and 0 ≺ x ≺ y then

∥x∥_{min} ≤ ∥y∥_{min}

where ∥x∥_{min} = ∥x∥_{B(H⊗_{2}\overline{H})}.

Proof. Let x = ∑ a_{k} ⊗ ā_{k}. Then the lemma is immediate from the identity

∥∑ a_{k} ⊗ ā_{k}∥ = sup {(∑ ∥ξa_{k}η∥_{2}^{2})^{1/2} | ∥ξ∥_{4} ≤ 1, ∥η∥_{4} ≤ 1}

for which we refer to [22]. Indeed, if d = ∑ b_{j} ⊗ b̄_{j} and y = x + d this identity applied to x + d makes it clear that ∥x∥ ≤ ∥x + d∥ = ∥y∥.

It will be convenient to extend our notation: Let H_{1}, H_{2}, …, H_{m} be an m-tuple of Hilbert spaces. For any k = 1, …, m we set

H_{m+k} = \overline{H}_{k}.

Let σ be any permutation of {1, …, 2m}. For any element x in B(H_{1}) ⊗ ⋯ ⊗ B(H_{2m}) we denote by

σ · x ∈ B(H_{σ(1)}) ⊗ ⋯ ⊗ B(H_{σ(2m)})

the element obtained from x by applying σ to the factors, i.e. if x = t_{1}⊗⋯⊗t_{2m} then σ · x = t_{σ(1)} ⊗ ⋯ ⊗ t_{σ(2m)} and x → σ · x is the linear extension of this map. Let H = H_{1} ⊗ ⋯ ⊗ H_{m}. Now if we are given a permutation σ and x, y in B(H_{σ(1)}) ⊗ ⋯ ⊗ B(H_{σ(2m)}) we note that

σ^{−1} · x, σ^{−1} · y ∈ B(H) ⊗ \overline{B(H)}.

We will write (abusively) x ≺ y (or y ≻ x) if we have

σ^{−1} · x ≺ σ^{−1} · y.

Of course this order depends on σ and although our notation does not keep track of that, we will need to remember σ, but hopefully no confusion should arise. While we will use various choices for σ, we never change our choice in the middle of a calculation, e.g. when adding two “positive” terms. For instance we allow ourselves to write that ∀a_{k} ∈ B(H) ∀b_{k} ∈ B(K) we have

∑ a_{k} ⊗ ā_{k} ⊗ b_{k} ⊗ b̄_{k} ≻ 0  (2.4)

in B(H ⊗ \overline{H} ⊗ K ⊗ \overline{K}), where implicitly we are referring to the permutation σ that takes H ⊗ \overline{H} ⊗ K ⊗ \overline{K} to H ⊗ K ⊗ \overline{H} ⊗ \overline{K}. In particular, we note that with this convention ∀x, y ∈ B(H) ⊗ \overline{B(H)} ∀b ∈ B(K) we have

x ≺ y ⇒ b ⊗ x ⊗ b̄ ≺ b ⊗ y ⊗ b̄.  (2.5)

Note that since the minimal tensor product is commutative, we still have, for any x, y in B(H_{σ(1)}) ⊗ ⋯ ⊗ B(H_{σ(2m)}), that

0 ≺ x ≺ y ⇒ ∥x∥_{min} ≤ ∥y∥_{min}.  (2.6)

From the obvious identity (x, y ∈ B(H))

(x + y) ⊗ \overline{(x + y)} + (x − y) ⊗ \overline{(x − y)} = 2(x ⊗ x̄ + y ⊗ ȳ)

it follows that

(x + y) ⊗ \overline{(x + y)} ≺ 2(x ⊗ x̄ + y ⊗ ȳ).  (2.7)

Note that if we set Φ(x) = x ⊗ x̄, then the preceding expresses the “order convexity” of this function:

Φ((x + y)/2) ≺ (Φ(x) + Φ(y))/2.

More generally, for any finite set x_{1}, …, x_{n} in B(H) we have Φ(n^{−1} ∑_{1}^{n} x_{k}) ≺ n^{−1} ∑_{1}^{n} Φ(x_{k}), or

(∑_{1}^{n} x_{k}) ⊗ \overline{(∑_{1}^{n} x_{k})} ≺ n ∑_{1}^{n} x_{k} ⊗ x̄_{k}.  (2.8)

We need to record below several variants of the “order convexity” of Φ. From now on we assume that H is a separable Hilbert space. More generally, for any x in B(H) ⊗ L_{2}(Ω, A, ℙ) and any σ-subalgebra B ⊂ A, we may associate to x ⊗ x̄ the function x ˙⊗x̄: ω → B(H) ⊗ \overline{B(H)} defined by x ˙⊗x̄(ω) = x(ω) ⊗ x̄(ω). We have then almost surely

0 ≺ 𝔼^{B}(x ˙⊗x̄),  (2.9)

and more precisely (again almost surely)

(𝔼^{B}x) ˙⊗\overline{(𝔼^{B}x)} ≺ 𝔼^{B}(x ˙⊗x̄).  (2.10)

Indeed, (2.9) follows from Lemma 2.1 (separable case) and the right hand side of (2.10) is equal to

(𝔼^{B}x) ˙⊗\overline{(𝔼^{B}x)} + 𝔼^{B}(y ˙⊗ȳ) where y = x − 𝔼^{B}x.

When B is the trivial algebra, we obtain

0 ≺ ∫ x ˙⊗x̄,  (2.11)

and hence for any measurable subset A ⊂ Ω

∫_{A} x ˙⊗x̄ dℙ ≺ ∫_{Ω} x ˙⊗x̄ dℙ,  (2.12)

and also

(𝔼x) ⊗ (𝔼x̄) ≺ 𝔼(x ˙⊗x̄).  (2.13)

We need to observe that for any integer m ≥ 1 we have

0 ≺ x ⇒ 0 ≺ x^{⊗m}  (2.14)

and more generally

0 ≺ x ≺ y ⇒ 0 ≺ x^{⊗m} ≺ y^{⊗m}.  (2.15)

Furthermore, if x_{1}, y_{1} ∈ B(H_{1}) ⊗ \overline{B(H_{1})} and x_{2}, y_{2} ∈ B(H_{2}) ⊗ \overline{B(H_{2})}

0 ≺ x_{1} ≺ y_{1} and 0 ≺ x_{2} ≺ y_{2} ⇒ 0 ≺ x_{1}⊗x_{2} ≺ y_{1}⊗y_{2},  (2.16)

where the natural permutation is applied to x_{1}⊗x_{2} and y_{1}⊗y_{2}, allowing to view them as elements of B(H_{1} ⊗ H_{2}) ⊗ \overline{B(H_{1} ⊗ H_{2})}. Returning to (2.10), we note that it implies

((𝔼^{B}x) ˙⊗\overline{(𝔼^{B}x)})^{⊗2} ≺ (𝔼^{B}(x ˙⊗x̄))^{⊗2} ≺ 𝔼^{B}(x ˙⊗x̄ ˙⊗x ˙⊗x̄),

and hence

∥((𝔼^{B}x) ˙⊗\overline{(𝔼^{B}x)})^{⊗2}∥ ≤ ∥𝔼^{B}(x ˙⊗x̄ ˙⊗x ˙⊗x̄)∥.

More generally, iterating this argument, we obtain for any integer k ≥ 1

((𝔼^{B}x) ˙⊗\overline{(𝔼^{B}x)})^{⊗2^{k}} ≺ 𝔼^{B}((x ˙⊗x̄)^{⊗2^{k}}).  (2.17)

In particular

((𝔼x) ˙⊗\overline{(𝔼x)})^{⊗2^{k}} ≺ 𝔼((x ˙⊗x̄)^{⊗2^{k}}),  (2.18)

and consequently for any finite sequence x_{1}, ⋯, x_{n} ∈ B(H) ⊗ L_{2}(Ω, A, ℙ)

∥ ∑_{j}((𝔼^{B}x_{j}) ˙⊗\overline{(𝔼^{B}x_{j})})^{⊗2^{k}}∥ ≤ ∥ ∑_{j} 𝔼^{B}((x_{j} ˙⊗x̄_{j})^{⊗2^{k}})∥.  (2.19)

In a somewhat different direction, for any measure μ, let f ∈ B(H) ⊗ L_{2}(μ), let P be any orthogonal projection on L_{2}(μ) and let g = (I ⊗ P)(f). Then

0 ≺ ∫ g ˙⊗ḡdμ ≺ ∫ f ˙⊗f̄dμ.  (2.20)

Indeed, this is immediate by (2.3).

3 Definition of Λ_{2m} Λ_{2m} Λ_{2m}

Our definition of the operator space Λ_{2m}(μ) is based on the case m = 1, i.e. on the operator Hilbert space OH, studied at length in [22]. The latter is based on the already mentioned Cauchy–Schwarz inequality due to Haagerup as follows: Let H, K be Hilbert spaces and let a_{k} ∈ B(H), b_{k} ∈ B(K)

∥∑ a_{k} ⊗ b_{k}∥ ≤ ∥∑ a_{k} ⊗ ā_{k}∥^{1/2} ∥∑ b_{k} ⊗ b̄_{k}∥^{1/2} .  (3.1)

This is usually stated with ∑ a_{k} ⊗ b̄_{k} on the left hand side, but since the right hand side is unchanged if we replace b_{k} by b̄_{k} we may write this as well. It will be convenient for our exposition to use the functional version of (3.1) as follows: For any Hilbert spaces H, K and any f ∈ B(H)⊗L_{2}(μ) and g ∈ B(K)⊗L_{2}(μ), we denote by f ˙⊗g the B(H) ⊗ B(K)-valued function defined by

(f ˙⊗g)(ω) = f(ω) ⊗ g(ω).

Of course, using the identity B(H) ⊗ \overline{B(H)} ≃ B(H) ⊗ B(\overline{H}), this extends the previously introduced notation for f ˙⊗f̄: Ω → B(H) ⊗ \overline{B(H)}.

Similarly, given n measurable functions f_{j}: Ω → B(H_{j}), we denote by f_{1} ˙⊗ ⋯ ˙⊗f_{n} the pointwise product viewed as a function with values in B(H_{1})⊗ ⋯ ⊗ B(H_{n}). Note that if f_{j} corresponds to an element in B(H_{j}) ⊗ L_{p_{j}} with p_{j} > 0 such that ∑ p_{j}^{−1} = p^{−1}, then by Hölder’s inequality, f_{1} ˙⊗ ⋯ ˙⊗f_{n} ∈ B(H_{1}) ⊗ ⋯ ⊗ B(H_{n}) ⊗ L_{p}. By (2.1) applied with H = L_{2}(μ), we have

∥∫ f ˙⊗ḡ dμ∥_{B(H⊗\overline{K})} ≤ ∥∫ f ˙⊗f̄ dμ∥_{B(H⊗\overline{H})}^{1/2} ∥∫ g ˙⊗ḡ dμ∥_{B(K⊗\overline{K})}^{1/2} . (3.2)

Replacing ḡ by g, we obtain

∥∫ f ˙⊗g dμ∥_{B(H⊗K)} ≤ ∥∫ f ˙⊗f̄ dμ∥_{B(H⊗\overline{H})}^{1/2} ∥∫ g ˙⊗ḡ dμ∥_{B(K⊗\overline{K})}^{1/2} . (3.3)

We note that this functional variant of (3.1) appears in unpublished work by Furman and Shalom (personal communication). We will also invoke the following variant: for any ψ ∈ B(ℓ_{2}) ⊗ L_{∞} with norm ∥ψ∥_{min} ≤ 1 we have

∥∫ f ˙⊗g ˙⊗ψ dμ∥_{B(H⊗K⊗ℓ_{2})} ≤ ∥∫ f ˙⊗f̄ dμ∥_{B(H⊗\overline{H})}^{1/2} ∥∫ g ˙⊗ḡ dμ∥_{B(K⊗\overline{K})}^{1/2} .

(3.4)

This (which can be interpreted as saying that the product map L_{2} × L_{2} → L_{1} is jointly completely contractive) can be verified rather easily using complex interpolation, see e.g. the proof of Lemma 7.1 below for a more detailed argument. For simplicity in this section we abbreviate L_{p}(μ) or L_{p}(Ω, μ) and we simply write L_{p} instead. We start by a version of Hölder’s inequality adapted to our needs that follows easily from (3.3). The proof uses an iteration idea already appearing in [6].

Lemma 3.1. Let m ≥ 1 be any integer. Then for any f_{1}, …, f_{2m} in B(H)⊗L_{2m} we have f_{1} ˙⊗ ⋯ ˙⊗f_{2m} ∈ B(H)^{⊗2m} ⊗ L_{1} and

∥∫ f_{1} ˙⊗ ⋯ ˙⊗f_{2m} dμ∥ ≤ ∏_{k=1}^{2m} ∥∫ f_{k}^{˙⊗m} ˙⊗f̄_{k}^{˙⊗m} dμ∥^{\frac{1}{2m}} ,  (3.5)

where we denote f^{˙⊗m} = f ˙⊗ ⋯ ˙⊗f (m times).

Proof. By homogeneity we may (and do) normalize and assume that

∥∫ f_{k}^{˙⊗m} ˙⊗f̄_{k}^{˙⊗m} dμ∥ ≤ 1.  ∀k = 1, …, 2m

Let

C = max {∥ ∫ g_{1} ˙⊗ ⋯ ˙⊗g_{2m} dμ∥}

where the maximum runs over all g_{k} in the set {f_{1}, …, f_{2m}, f̄_{1}, …, f̄_{2m}}. It clearly suffices to prove C ≤ 1. In the interest of the reader, we first do the proof in the simplest case m = 2. We have by (3.3)

∥ ∫ f_{1} ˙⊗ ⋯ ˙⊗f_{4}dμ∥ ≤ ∥ ∫ f_{1} ˙⊗f_{2} ˙⊗\overline{f_{1} ˙⊗f_{2}}dμ∥^{1/2}∥ ∫ f_{3} ˙⊗f_{4} ˙⊗\overline{f_{3} ˙⊗f_{4}}dμ∥^{1/2}

which we may rewrite as

∥ ∫ f_{1} ˙⊗ ⋯ ˙⊗f_{4}dμ∥ ≤ ∥ ∫ f_{1} ˙⊗f̄_{1} ˙⊗f_{2} ˙⊗f̄_{2}dμ∥^{1/2}∥ ∫ f_{3} ˙⊗f̄_{3} ˙⊗f_{4} ˙⊗f̄_{4}dμ∥^{1/2},

(3.6)

and by (3.3) again we have

∥ ∫ f_{1} ˙⊗f̄_{1} ˙⊗f_{2} ˙⊗f̄_{2}dμ∥^{1/2} ≤

≤ ∥ ∫ f_{1} ˙⊗f̄_{1} ˙⊗f̄_{1} ˙⊗f_{1}dμ∥^{1/2}∥ ∫ f_{2} ˙⊗f̄_{2} ˙⊗f̄_{2} ˙⊗f_{2}dμ∥^{1/2} ≤ 1

and similarly for the other factor in (3.6). Thus we obtain the announced inequality for m = 2. To check the general case, let us denote

I(f_{1}, …, f_{2m}) = ∥∫ f_{1} ˙⊗ ⋯ ˙⊗f_{2m} dμ∥ .

By (3.3) we find

I(f_{1}, …, f_{2m}) ≤ (I(f_{1}, …, f_{m}, f̄_{1}, …, f̄_{m})C)^{1/2}.

Note that I(f_{1}, …, f_{2m}) is invariant under permutation of entries. Thus we have

I(f_{1}, …, f_{m}, f̄_{1}, …, f̄_{m}) = I(f_{1}, f̄_{1}, f_{2}, f̄_{2}, …, f_{m}, f̄_{m}).

Using (3.3) again we find

I(f_{1}, …, f_{m}, f̄_{1}, …, f̄_{m}) ≤ (I(f_{1}, f̄_{1}, f_{1}, f̄_{1}, f_{2}, f̄_{2}, …)C)^{1/2}

and continuing in this way we obtain

I(f_{1}, …, f_{2m}) ≤ I(f_{1}, f̄_{1}, f_{1}, f̄_{1}, …, f_{1}, f̄_{1})^{θ}C^{1−θ}

where 0 < θ < 1 is equal to 2^{−K} with K the number of iterations. Since we assume I(f_{1}, f̄_{1}, …, f_{1}, f̄_{1}) ≤ 1 we find

I(f_{1}, …, f_{2m}) ≤ C^{1−θ}.

But we may replace f_{1}, …, f_{2m} by g_{1}, …, g_{2m} and the same argument gives us

I(g_{1}, …, g_{2m}) ≤ C^{1−θ}

and hence C ≤ C^{1−θ}, from which C ≤ 1 follows immediately.

Proposition 3.2. Let m ≥ 1 be an integer and p = 2m. There is an isometric embedding

L_{p}(μ) ⊂ B(H)

so that for any f ∈ B(H) ⊗ L_{p}(μ) and any H we have

∥f∥_{B(H⊗H)} = ∥∫ f^{˙⊗m} ˙⊗f̄^{˙⊗m} dμ∥^{\frac{1}{2m}}

where we again denote f^{˙⊗m} = f ˙⊗ ⋯ ˙⊗f (m times).

Proof. Let B = B(H). With the notation in Lemma 3.1 we have

∥∫ f^{˙⊗m} ˙⊗f̄^{˙⊗m} dμ∥^{\frac{1}{2m}} = max\{I(f, f̄, g_{2}, ḡ_{2}, …, g_{m}, ḡ_{m})^{1/2}\}  (3.7)

where the supremum runs over all g_{2}, …, g_{m} in B ⊗ L_{2m} such that ∥ ∫ g_{k}^{˙⊗m} ˙⊗ḡ_{k}^{˙⊗m} dμ∥ ≤ 1 for any k = 1, …, m. Indeed, it follows easily from (3.5) that the latter maximum is attained for the choice of g_{2} = ⋯ = g_{m} = λf with λ = ∥ ∫ f^{˙⊗m} ˙⊗f̄^{˙⊗m} dμ∥^{−1/2m}. Thus we can proceed as in [22] for the case m = 1: We assume H = ℓ_{2} to fix ideas. Let S denote the collection of all G = g_{2} ˙⊗ ⋯ ˙⊗g_{m} where (g_{2}, …, g_{m}) runs over the set appearing in (3.7). Note that by (3.5) we know that for any f in B ⊗ L_{p} we have f ˙⊗G ∈ B(H^{⊗m}) ⊗ L_{2}. Then for any G in S we introduce the linear map

u_{G}: L_{p}(μ) → B(H^{⊗m}) ⊗_{min} (L_{2})_{oh}

defined by

u_{G}(f) = f ˙⊗G.

Then (3.5) and (1.4) imply ∥u_{G}∥ ≤ 1. We then define the embedding

u: L_{p}(μ) → ⨁_{G∈S} B(H^{⊗m−1}) ⊗_{min} (L_{2})_{oh}

by setting

u(f) = ⨁_{G} u_{G}(f).

Then (3.7) gives us that for any f in B ⊗ L_{p} we have

∥(id ⊗ u)(f)∥ = ∥∫ f^{˙⊗m} ˙⊗f̄^{˙⊗m} dμ∥^{\frac{1}{2m}} .

Thus the embedding u has the required properties.

Definition 3.3. We denote by Λ_{p}(μ) (p = 2m) the operator space appearing in Proposition 3.2. Note that Λ_{p}(μ) is isometric to L_{p}(μ) and for any H and for any f ∈ B(H) ⊗ Λ_{p}(μ) we have

∥f∥_{B(H)⊗_{min}Λ_{p}(μ)} = ∥∫ f^{˙⊗m} ˙⊗f̄^{˙⊗m} dμ∥^{\frac{1}{2m}} .  (3.8)

Proposition 3.4. Let f be as in Definition 3.3 with p = 2m. Then

∥f∥_{B(H)⊗_{min}Λ_{p}} = ∥f^{˙⊗m}∥_{B(H^{⊗m})⊗_{min}L_{2}}^{1/m} = ∥f^{˙⊗m} ˙⊗f̄^{˙⊗m}∥B(H^{⊗m}⊗H̄^{⊗m})⊗_{min}L_{1}^{\frac{1}{2m}}

(3.9)

where L_{2} and L_{1} are equipped respectively with their oh and maximal operator space structure. So if we make the convention that Λ_{1} = L_{1} equipped with its maximal o.s.s. then we have

∥f∥_{B(H)⊗_{min}Λ_{2m}} = ∥(f ˙⊗f̄)^{˙⊗m}∥B((H⊗H̄)^{⊗m})⊗_{min}Λ_{1}^{\frac{1}{2m}}.

Proof. The first equality is immediate since it is easy to check that for any g ∈ B(H) ⊗ L_{2} we have

∥g∥_{min} = ∥∫ g ˙⊗ḡ dμ∥_{B(H⊗\overline{H})}^{1/2} .

For the second one, we use (1.2): for any φ in B(H) ⊗ L_{1}

∥φ∥_{B(H)⊗_{min}L_{1}} = sup {∥∫ φ ˙⊗ψ dμ∥}

where the supremum runs over all ψ in B(ℓ_{2})⊗L_{∞} with ∥ψ∥_{min} ≤ 1. Applying this to φ(t) = f(t)^{⊗m} ⊗ \overline{f(t)}^{⊗m} we find using (3.4)

∥f^{⊗m} ⊗ f̄^{⊗m}∥_{B(H)⊗_{min}L_{1}} ≤ ∥f∥_{B(H)⊗_{min}Λ_{p}}^{2m},

and the choice of ψ ≡ 1 shows that this inequality is actually an equality.

Notation: For any f ∈ B(H) ⊗ L_{p}(μ) (or equivalently f ∈ B(H) ⊗ Λ_{p}(μ)), we denote

∥f∥_{(p)} = ∥f∥_{B(H)⊗_{min}Λ_{p}(μ)}.

The preceding Proposition shows that if p = 2m, we have

∥f∥_{(p)} = ∥f^{˙⊗p/2}∥_{(2)}^{2/p} = ∥(f ˙⊗f̄)^{˙⊗p/2}∥_{(1)}^{1/p}.

Note that by (2.15), if g, f ∈ B(H) ⊗ L_{p}(μ) and p = 2m

(g(ω) ˙⊗ḡ(ω) ≺ f(ω) ˙⊗f̄(ω) ∀_{a.s} ω ∈ Ω) ⇒ ∥g∥_{(p)} ≤ ∥f∥_{(p)}.  (3.10)

Corollary 3.5. The conditional expectation 𝔼^{B} is completely contractive on Λ_{p} for any p ∈ 2ℕ.

Proof. The case p = 2 is clear by the homogeneity of the spaces OH. Using (2.10) one easily passes from p to 2p. By induction, this proves the Corollary for any p of the form p = 2^{m}. For p equal to an arbitrary even integer, we need a different argument. Let f ∈ B(H) ⊗ Λ_{p} and g = 𝔼^{B}f. By the classical property of conditional expectations, we have

∥g∥_{(p)}^{p} = ∥ ∫ g ˙⊗ḡ ˙⊗g ˙⊗ḡ ⋯ dμ∥ = ∥ ∫ f ˙⊗ḡ ˙⊗g ˙⊗ḡ ⋯ dμ∥

and hence by (3.5) ∥g∥_{(p)}^{p} ≤ ∥f∥_{(p)}∥g∥_{(p)}^{p−1} or equivalently ∥g∥_{(p)} ≤ ∥f∥_{(p)}.

With the above definition, Lemma 3.1 together with (3.4) immediately implies (see the beginning of §7 for the definition of jointly completely contractive multilinear maps):

Corollary 3.6. Let p = 2m. The product mapping (f_{1}, …, f_{2m}) ↦ f_{1} × … × f_{2m} is a jointly completely contractive multilinear map from Λ_{p}(μ)^{2m} to Λ_{1}(μ) = L_{1}(μ).

Proof. By Lemma 3.1 and (3.4) we have for any ψ ∈ B(ℓ_{2}) ⊗ L_{∞} with norm ∥ψ∥_{min} ≤ 1

∥ ∫ f_{1} ˙⊗ ⋯ ˙⊗f_{p} ˙⊗ψ∥ ≤ ∏_{1}^{p} ∥f_{j}∥_{(p)}.

Taking the supremum over all such ψ we obtain

∥f_{1} ˙⊗ ⋯ ˙⊗f_{p}∥_{(1)} ≤ ∏_{1}^{p} ∥f_{j}∥_{(p)},

which is nothing but a reformulation of the assertion of this corollary.

Corollary 3.7. Let p ≥ q ≥ 2 be even integers. If μ is a probability, the inclusion L_{p}(μ) ⊂ L_{q}(μ) is a complete contraction from Λ_{p}(μ) to Λ_{q}(μ).

Proof. Take p = 2m, q = 2n and r = 2m − 2n with n < m. Let f ∈ B(H) ⊗ L_{p}(μ) and g ∈ B(H)⊗L_{p}(μ). Consider f_{1} ⊗⋯⊗f_{2m} = (f ⊗f̄)^{n} ⊗g^{⊗r}, and let us choose for g the constant function that is identically equal to the identity operator on H. Then, using (3.8), (3.5) yields

∥f∥_{B(H)⊗_{min}Λ_{q}(μ)}^{q} ≤ ∥f∥_{B(H)⊗_{min}Λ_{p}(μ)}^{q} × 1^{p−q}

and hence ∥f∥_{B(H)⊗_{min}Λ_{q}(μ)} ≤ ∥f∥_{B(H)⊗_{min}Λ_{p}(μ)}.

Remark 3.8. Let f ∈ B(H)⊗L_{∞}(μ). We may view f as an essentially bounded B(H)-valued function. Then if μ is finite we have

∥f∥_{L_{∞}(μ;B(H))} = lim_{p→∞} ∥f∥_{(p)}.  (3.11)

Indeed, we may assume μ(Ω) = 1 and it clearly suffices to prove that for any measurable subset A ⊂ Ω with μ(A) > 0 we have

∥μ(A)^{−1} ∫_{A} f dμ∥ ≤ lim_{p→∞} ∥f∥_{(p)}.  (3.12)

To verify this let f_{A} = μ(A)^{−1} ∫_{A} f dμ. By (2.17) with p = 2^{k} we have

∥f_{A}∥ = ∥(f_{A} ⊗ f̄_{A})^{⊗p}∥^{\frac{1}{2p}} ≤ ∥μ(A)^{−1} ∫_{A} (f ˙⊗f̄)^{˙⊗p}∥^{\frac{1}{2p}}

≤ μ(A)^{− \frac{1}{2p}} ∥f∥_{(2p)}

where we also use (2.12) and Lemma 2.3, and hence letting p → ∞ we obtain (3.12).

4 Martingale inequalities in Λ_{p} Λ_{p} Λ_{p}

Our main result is the following one. This is an operator valued version of Burkholder’s martingale inequalities. Although our inequality seems very different from the one appearing in [26], the method used to prove it is rather similar. Throughout this section H is an arbitrary Hilbert space and we set B = B(H). We give ourselves a probability space (Ω, A, ℙ) and we set L_{p} = L_{p}(Ω, A, ℙ). We also give ourselves a filtration (A_{n})_{n≥0} on (Ω, A, ℙ). We assume that A_{0} is trivial and that A_{∞} = σ( ⋃_{n≥0} A_{n}) is equal to A. We may view any f in B ⊗ L_{p} as a B-valued random variable f: Ω → B. We denote 𝔼_{n}f = 𝔼^{A_{n}}f, d_{0} = 𝔼_{0}f and d_{n} = 𝔼_{n}f − 𝔼_{n−1}f for any n ≥ 1. We will say that f ∈ B ⊗L_{p} is a test function if it is A_{N}-measurable for some N ≥ 1, or equivalently if f can be written as a finite sum f = ∑_{0}^{∞} d_{n}. We then denote

S(f) = ∑_{0}^{∞} d_{n} ˙⊗d̄_{n}.

We also denote

σ(f) = d_{0} ˙⊗d̄_{0} + ∑_{1}^{∞} 𝔼_{n−1}(d_{n} ˙⊗d̄_{n}).

We recall the notation

x^{˙⊗m} = x ˙⊗ ⋯ ˙⊗x

where x is repeated m times. Note that σ(f) ∈ B ⊗ \overline{B} ⊗ L_{p/2} and also that

(d_{n} ˙⊗d̄_{n})^{˙⊗p/2} ∈ (B ⊗ \overline{B})^{⊗p/2} ⊗ L_{1}.

Theorem 4.1. For any p ≥ 2 of the form p = 2^{k} for some k ≥ 1 there are positive constants C_{1}(p), C_{2}(p) such that for any test function f in B ⊗ L_{p} we have

C_{1}(p)^{−1}∥S(f)∥_{B⊗_{min}\overline{B}⊗_{min}Λ_{p/2}}^{1/2} ≤ ∥f∥_{B⊗_{min}Λ_{p}} ≤ C_{2}(p)∥S(f)∥_{B⊗_{min}\overline{B}⊗_{min}Λ_{p/2}}^{1/2}.

(4.1)

First part of the proof. We start by the case p = 4. Let f = ∑ d_{n}, S = S(f) and σ = σ(f). Then

f ˙⊗f̄ = S + a + b

where a = ∑ d_{n} ˙⊗f̄_{n−1} and b = ∑ f_{n−1} ˙⊗d̄_{n}. Let g = a+b so that f ˙⊗f̄−S = g. Note that by (2.7) applied pointwise (or, say, alomst surely)

g ˙⊗ḡ ≺ 2(a ˙⊗ā + b ˙⊗b̄)

and hence 𝔼(g ˙⊗ḡ) ≺ 2𝔼(a ˙⊗ā) + 2𝔼(b ˙⊗b̄). By Lemma 2.3 we have

∥𝔼(g ˙⊗ḡ)∥ ≤ 2∥𝔼(a ˙⊗ā)∥ + 2𝔼(b ˙⊗b̄)∥.  (4.2)

Note that for any f

∥𝔼(f ˙⊗f̄)∥^{1/2} = ∥f∥_{B⊗_{min}Λ_{2}}.

By orthogonality 𝔼(a ˙⊗ā) = 𝔼(∑ d_{n} ˙⊗f̄_{n−1} ˙⊗d̄_{n} ˙⊗f_{n−1}). By (2.10) we have

f̄_{n−1} ˙⊗f_{n−1} ≺ 𝔼_{n−1}(f̄ ˙⊗f)

therefore by (2.5)

d_{n} ˙⊗d̄_{n} ˙⊗f̄_{n−1} ˙⊗f_{n−1} ≺ d_{n} ˙⊗d̄_{n} ˙⊗𝔼_{n−1}(f̄ ˙⊗f)

and hence

𝔼(d_{n} ˙⊗d̄_{n} ˙⊗f̄_{n−1} ˙⊗f_{n−1}) ≺ 𝔼(𝔼_{n−1}(d_{n} ˙⊗d̄_{n}) ˙⊗f̄ ˙⊗f)

which yields (after a suitable permutation) by Lemma 2.3

∥𝔼(a ˙⊗ā)∥ ≤ ∥𝔼(σ ˙⊗f̄ ˙⊗f)∥,

and hence by Haagerup’s Cauchy–Schwarz inequality

∥𝔼(a ˙⊗ā)∥ ≤ ∥𝔼(σ ˙⊗σ̄)∥^{1/2}∥𝔼(f̄ ˙⊗f ˙⊗f ˙⊗f̄)∥^{1/2}.  (4.3)

Similarly we find

∥𝔼(b ˙⊗b̄)∥ ≤ ∥𝔼(σ ˙⊗σ̄)∥^{1/2}∥𝔼(f ˙⊗f̄ ˙⊗f̄ ˙⊗f)∥^{1/2}.  (4.4)

We now claim that

∥𝔼(σ ˙⊗σ̄)∥^{1/2} ≤ 2∥𝔼(S ˙⊗\overline{S})∥^{1/2}.

Using this claim the conclusion is easy: By (4.2), (4.3), (4.4) we have

∥𝔼(g ˙⊗ḡ)∥ ≤ 8∥𝔼(S ˙⊗\overline{S})∥^{1/2}∥𝔼(f ˙⊗f̄ ˙⊗f ˙⊗f̄)∥^{1/2}.

But now g → ∥𝔼(g ˙⊗ḡ)∥^{1/2} = ∥g∥_{B⊗_{min}L_{2}} is a norm so by its subadditivity, recalling g = f ˙⊗f̄− S, we have

|∥f ˙⊗f̄∥_{B⊗_{min}L_{2}} − ∥S∥_{B⊗_{min}L_{2}} | ≤ ∥g∥_{B⊗_{min}L_{2}} ≤ 8^{1/2}xy

where x^{2} = ∥f ˙⊗f̄∥_{B_{min}⊗L_{2}} and y^{2} = ∥S∥_{B⊗_{min}L_{2}}. Equivalently, we have

|x^{2} − y^{2}| ≤ 8^{1/2}xy

from which it immediately follows that

max \{\frac{x}{y}, \frac{y}{x}\} ≤ \sqrt{2} + \sqrt{3}.

Thus, modulo our claim, we obtain the announced inequality (4.1) for p = 4. To prove the claim we note that it is a particular case of the “dual Doob inequality” appearing in the next Lemma.

Lemma 4.2. Let θ_{1}, …, θ_{N} be arbitrary in B ⊗ L_{4}. Let α = ∑_{1}^{N} 𝔼_{n}(θ_{n} ˙⊗θ̄_{n}) and β = ∑_{1}^{N} θ_{n} ˙⊗θ̄_{n}. Then

∥𝔼(α ˙⊗ᾱ)∥^{1/2} ≤ 2∥𝔼(β ˙⊗β̄)∥^{1/2}.

Proof. Let α_{n} = 𝔼_{n}(θ_{n} ˙⊗θ̄_{n}) and β_{n} = θ_{n} ˙⊗θ̄_{n}. Note that β_{n} ≻ 0 and α_{n} ≻ 0. We have α ˙⊗ᾱ = ∑_{n,k} α_{n} ˙⊗ᾱ_{k} and hence

𝔼(α ˙⊗ᾱ) = 𝔼 (∑_{n≤k} 𝔼_{n}β_{n} ˙⊗\overline{𝔼_{k}β_{k}}) + 𝔼 (∑_{n>k} 𝔼_{n}β_{n} ˙⊗\overline{𝔼_{k}β_{k}})

= 𝔼 (∑_{n≤k}(𝔼_{n}β_{n}) ˙⊗β̄_{k}) + 𝔼 (∑_{n>k} β_{n} ˙⊗(\overline{𝔼_{k}β_{k}}))

= I + II.

Using a suitable permutation (as explained before (2.5)) we have by (2.5) or (2.16)

(𝔼_{n}β_{n}) ˙⊗β̄_{k} ≻ 0 and β_{n} ˙⊗(\overline{𝔼_{k}β_{k}}) ≻ 0.

Therefore, using (2.9), it follows from Lemma 2.3 that

∥I∥ ≤ ∥∑_{n,k} 𝔼((𝔼_{n}β_{n}) ˙⊗β̄_{k})∥ = ∥𝔼(α ˙⊗β̄)∥,

and hence by (3.3)

∥I∥ ≤ ∥𝔼(α ˙⊗ᾱ)∥^{1/2}∥𝔼(β ˙⊗β̄)∥^{1/2}.

A similar bound holds for ∥II∥. Thus we obtain

∥𝔼(α ˙⊗ᾱ)∥ ≤ ∥I∥ + ∥II∥ ≤ 2∥𝔼(α ˙⊗ᾱ)∥^{1/2}∥𝔼(β ˙⊗β̄)∥^{1/2}.

After division by ∥𝔼(α ˙⊗ᾱ)∥^{1/2} we find the inequality in Lemma 4.2.

We will need to extend Lemma 4.2 as follows:

Lemma 4.3. Let m ≥ 1 be any integer. Let θ_{1}, …, θ_{N} be arbitrary in B ⊗ L_{2m}. Let α, β be as in the preceding lemma. Then

∥𝔼(α^{˙⊗m})∥ ≤ m^{m}∥𝔼(β^{˙⊗m})∥.  (4.5)

Proof. Note that up to a permutation of factors (α)^{˙⊗2} and α ˙⊗ᾱ are the same. In any case ∥𝔼((α)^{˙⊗2})∥ = ∥𝔼(α ˙⊗ᾱ)∥, so the case m = 2 follows from the preceding lemma (and m = 1 is trivial). We use the same notation as in the preceding proof. We can write

(α)^{˙⊗m} = ∑_{n(1),...,n(m)} α_{n(1)} ˙⊗α_{n(2)} ˙⊗ ⋯ ˙⊗α_{n(m)}.

We can partition the set of m-tuples n = (n(1), …, n(m)) into subsets S_{1}, …, S_{m} so that we have

n(1) = max_{j} n(j),  ∀n ∈ S_{1}
n(2) = max_{j} n(j),  ∀n ∈ S_{2}

and so on. Let then T (j) = 𝔼 (∑_{n∈S_{j}} α_{n(1)} ˙⊗ ⋯ ˙⊗ᾱ_{n(m)}). Consider j = 1 for simplicity, arguing as in the preceding proof, we have

T (1) = ∑_{n∈S_{1}} 𝔼(α_{n(1)} ˙⊗ ⋯ ˙⊗α_{n(m)})

= ∑_{n∈S_{1}} 𝔼(β_{n(1)} ˙⊗α_{n(2)} ⋯ ˙⊗α_{n(m)})

≺ ∑_{n(1),...,n(m)} 𝔼(β_{n(1)} ˙⊗α_{n(2)} ⋯ ˙⊗α_{n(m)}) = 𝔼(β ˙⊗α^{˙⊗m−1})

and hence by (3.5)

∥𝔼T (1)∥ ≤ ∥𝔼(β ˙⊗α^{˙⊗m−1})∥ ≤ ∥𝔼(β^{˙⊗m})∥^{\frac{1}{m}} ∥𝔼(α^{˙⊗m})∥^{\frac{m−1}{m}} .

A similar bound holds for each T (j) (j = 1, …, m). Thus we find

∥𝔼(α^{˙⊗m})∥ = ∥∑_{1}^{m} 𝔼T (j)∥ ≤ m∥𝔼(β^{˙⊗m})∥^{\frac{1}{m}} ∥𝔼(α^{˙⊗m}∥^{\frac{m−1}{m}} ,

and again after a suitable division we obtain (4.5).

Proof of Theorem 4.1 in the dyadic case. Assume that d_{n} ˙⊗d̄_{n} is A_{n−1}- measurable. Note that this holds in the dyadic case when Ω = {−1, 1}^{ℕ} as well as for the filtration naturally associated to the Haar orthonormal system. In that case we can give a short proof of the following inequality for any m ≥ 1

∥𝔼(S(a)^{˙⊗m})∥^{\frac{1}{m}} ≤ m∥𝔼(S^{˙⊗2m})∥^{\frac{1}{2m}} ∥𝔼((f ˙⊗f̄)^{˙⊗2m})∥^{\frac{1}{2m}} ,  (4.6)

and a similar bound for ∥𝔼(S(b)^{˙⊗m})∥^{\frac{1}{m}}. Indeed, recall a = ∑ d_{n} ˙⊗f̄_{n−1}. Note that S(a) is up to permutation the same as

S_{1} = ∑ d_{n} ˙⊗f̄_{n−1} ˙⊗f_{n−1} ˙⊗d̄_{n}.

By (2.10) and (2.5), we have

S_{1} ≺ ∑ d_{n} ˙⊗𝔼_{n−1}(f̄ ˙⊗f) ˙⊗d̄_{n} = ∑ 𝔼_{n−1}(d_{n} ˙⊗f̄ ˙⊗f ˙⊗d̄_{n})

where the last equality holds because d_{n} ˙⊗d̄_{n} is assumed (n − 1)-measurable. By (2.15) this implies

S_{1}^{˙⊗m} ≺ (∑ 𝔼_{n−1}(d_{n} ˙⊗f̄ ˙⊗f ˙⊗d̄_{n}))^{˙⊗m}

and hence by (4.5) (recall S = S(f) and the permutation invariance of the norm)

∥𝔼(S_{1}^{˙⊗m})∥ ≤ m^{m} ∥𝔼 ((∑ d_{n} ˙⊗f̄ ˙⊗f ˙⊗d̄_{n})^{˙⊗m})∥
= m^{m}∥𝔼((S ˙⊗f̄ ˙⊗f)^{˙⊗m})∥

and hence by Lemma 3.1

≤ m^{m}∥𝔼(S^{˙⊗2m})∥^{1/2}∥𝔼((f ˙⊗f̄)^{˙⊗2m})∥^{1/2}.

Thus we obtain (4.6) as announced:

∥𝔼(S(a)^{˙⊗m})∥ = ∥𝔼(S_{1}^{˙⊗m})∥ ≤ m^{m}∥𝔼(S^{˙⊗2m})∥^{1/2}∥𝔼((f ˙⊗f̄)^{˙⊗2m})∥^{1/2}.

The proof with b in place of a is identical. Using (4.6) and the analogue for b, it is easy to show (assuming that d_{n} ˙⊗d̄_{n} is A_{n−1}-measurable) that the validity of (4.1) for p = 2m implies its validity for p = 4m. Indeed, let us assume (4.1) for p = 2m and let C = C_{2}(2m). We find, recalling f ˙⊗f̄− S = a + b

∥f ˙⊗f̄− S∥_{B⊗_{min}Λ_{2m}} ≤ ∥a∥_{B⊗_{min}Λ_{2m}} + ∥b∥_{B⊗_{min}Λ_{2m}}
≤ C(∥S(a)∥_{B⊗_{min}Λ_{m}}^{1/2} + ∥S(b)∥_{B⊗_{min}Λ_{m}}^{1/2})

and hence by (4.6)

≤ 2Cm^{1/2}∥S∥_{B⊗_{min}Λ_{2m}}^{1/2}∥f ˙⊗f̄∥_{B⊗_{min}Λ_{2m}}^{1/2}.

Therefore, again setting x = ∥f ˙⊗f̄∥_{B⊗_{min}Λ_{2m}} and y = ∥S∥_{B⊗_{min}Λ_{2m}} we find

|x − y| ≤ 2Cm^{1/2}\sqrt{xy},

and we conclude as before that x and y are comparable, so that (4.1) holds for p = 4m.

The next corollary is now immediate from the dyadic case. However, we will later show that it is valid for any p in 2ℕ (see Corollary 11.3).

Corollary 4.4. Assume p ≥ 2 of the form p = 2^{k} for some k ≥ 1. If Ω = {−1, +1}^{ℕ}, the closed span of the coordinates (ε_{n}) (or equivalently, of the Rademacher functions on Ω = [0, 1]) in Λ_{p} is completely isomorphic to the space OH, i.e. to ℓ_{2} equipped with the o.s.s. of OH. Moreover, the orthogonal projection P onto it is c.b. on Λ_{p}.

Proof. Let f = ∑ x_{n}ε_{n} (x_{n} ∈ B(H)). By (4.1) , ∥f∥_{(p)} is equivalent to ∥ ∑ x_{n} ⊗x̄_{n}∥^{1/2} and the latter is equal to the norm of ∑ e_{n} ⊗x_{n} in OH ⊗_{min} B where (e_{n}) is any orthonormal basis of OH. Thus the closed span of (ε_{n}) in Λ_{p} is isomorphic to OH. We skip the proof of the complementation because we give the details for that in the proof of Proposition 11.1 below.

Proof of the right hand side of (4.1). We will use induction on k. The case k = 1 is clear. Assume that the right hand side of (4.1) holds for p = m, we will show it for p = 2m. With the preceding notation, recall g = a + b and hence our assumption yields

∥g∥_{B⊗\overline{B}⊗Λ_{m}} ≤ ∥a∥_{B⊗\overline{B}⊗Λ_{m}} + ∥b∥_{B⊗\overline{B}⊗Λ_{m}} ≤ C_{2}(m)(∥S(a)∥_{•}^{1/2} + ∥S(b)∥_{•}^{1/2})

(4.7)

where the dot stands for B ⊗_{min} \overline{B} ⊗_{min} \overline{B} ⊗_{min} B ⊗_{min} Λ_{m/2}. Since by (2.7)

f̄_{n−1} ˙⊗f_{n−1} ≺ 2(f̄_{n} ˙⊗f_{n} + d̄_{n} ˙⊗d_{n})

we have using (2.5) and (2.6) (in a suitable permutation)

∥S(a)∥_{•}^{1/2} = ∥∑ d_{n} ˙⊗f̄_{n−1} ˙⊗d̄_{n} ˙⊗f_{n−1}∥_{•}^{1/2} ≤ I + II

where

2^{−1/2}I = ∥∑ d_{n} ˙⊗f̄_{n} ˙⊗d̄_{n} ˙⊗f_{n}∥•^{\frac{1}{2}} and 2^{−1/2}II = ∥∑ d_{n} ˙⊗d̄_{n} ˙⊗d̄_{n} ˙⊗d_{n}∥•^{\frac{1}{2}} .

Note that for any F in B ⊗ \overline{B} ⊗ \overline{B} ⊗ B ⊗ Λ_{m/2} we have ∥F ∥_{•} = ∥𝔼((F ˙⊗\overline{F})^{˙⊗m/4})∥^{\frac{2}{m}}.

Recall that, by (2.10), f_{n} ˙⊗f̄_{n} ≺ 𝔼_{n}(f ˙⊗f̄). Thus we have by (2.5) (2.6) and (2.15)

2^{−1/2}I ≤ ∥𝔼 ((∑ d_{n} ˙⊗d̄_{n} ˙⊗𝔼_{n}(f ˙⊗f̄))^{˙⊗m/2})∥^{\frac{1}{m}}

= ∥𝔼 ((∑ 𝔼_{n}(d_{n} ˙⊗d̄_{n} ˙⊗f ˙⊗f̄))^{˙⊗m/2})∥^{\frac{1}{m}}

and hence by (4.5) and (3.5) (or actually (3.3))

≤ (m/2)^{1/2}∥𝔼((S ˙⊗f ˙⊗f̄)^{˙⊗m/2})∥^{\frac{1}{m}}
≤ (m/2)^{1/2}∥𝔼(S^{˙⊗m})∥^{\frac{1}{2m}} ∥𝔼((f ˙⊗f̄)^{˙⊗m})∥^{\frac{1}{2m}} .

Moreover, recalling (2.4), we have obviously 0 ≺ d_{n} ⊗ d̄_{n} ⊗ d̄_{k} ⊗ d_{k} for all n, k and hence ∑ d_{n} ˙⊗d̄_{n} ˙⊗d̄_{n} ˙⊗d_{n} ≺ S ˙⊗\overline{S}. Therefore, again by (2.6) and (2.15)

2^{−1/2}II ≤ ∥𝔼((S ˙⊗\overline{S} ˙⊗\overline{S} ˙⊗S)^{m/4})∥^{1/m} = ∥𝔼(S^{˙⊗m})∥^{\frac{1}{m}} .

Let x = ∥𝔼(S^{˙⊗m})∥^{\frac{1}{m}} and y = ∥𝔼(f ˙⊗f̄)^{˙⊗m}∥^{\frac{1}{m}}. This yields

∥S(a)∥•^{\frac{1}{2}} ≤ \sqrt{m}\sqrt{xy} + \sqrt{2}x

and a similar bound for S(b). Thus we obtain

∥g∥_{B⊗\overline{B}⊗Λ_{m}} ≤ 2C_{2}(m)(\sqrt{m}\sqrt{xy} + \sqrt{2}x).

Since g = f ˙⊗f̄− S we have

|∥f ˙⊗f̄∥_{B⊗\overline{B}⊗Λ_{m}} − ∥S∥_{B⊗\overline{B}⊗Λ_{m}} | ≤ ∥g∥_{B⊗\overline{B}⊗Λ_{m}}

and hence we obtain

|y − x| ≤ 2C_{2}(m)(\sqrt{m}\sqrt{xy} + \sqrt{2}x)

From the latter it is clear that there is a constant C_{2}(2m) such that

\sqrt{y} ≤ C_{2}(2m)\sqrt{x}

and this is the right hand side of (4.1) for p = 2m,

To prove the general case of both sides of (4.1), the following Lemma will be crucial. We will use this only for m = 1, but the inductive argument curiously requires to prove it for all dyadic m.

Lemma 4.5. Let f ∈ B(H)⊗L_{4mp} be a test function. As before we set f_{n} = 𝔼_{n}f and d_{n} = f_{n} − f_{n−1} for all n ≥ 1. Let p = 2^{k} for some integer k ≥ 0. Then, for any integer m ≥ 1 of the form m = 2^{ℓ} for some ℓ ≥ 0, there is a constant C = C(m, p) such that

∥∑_{1}^{∞}(d_{n} ˙⊗d̄_{n})^{˙⊗m} ˙⊗(f_{n−1} ˙⊗f̄_{n−1})^{˙⊗m}∥_{(p)} ≤ C∥S∥_{(2mp)}^{m}∥f∥_{(4mp)}^{2m}.  (4.8)

Proof. We use induction on k starting from p = 1. We may assume d_{0} = 0 for simplicity. Let

I(m, p) = ∥ ∑_{1}^{∞}(d_{n} ˙⊗d̄_{n})^{˙⊗m} ˙⊗(f_{n−1} ˙⊗f̄_{n−1})^{˙⊗m}∥_{(p)}.

By (2.17) and (2.5) (and by the self-adjointness of 𝔼_{n−1}) we have

I(m, 1) ≤ ∥𝔼 (∑_{1}^{∞}(d_{n} ˙⊗d̄_{n})^{˙⊗m} ˙⊗𝔼_{n−1}((f ˙⊗f̄)^{˙⊗m}))∥
= ∥𝔼 (∑_{1}^{∞} 𝔼_{n−1}((d_{n} ˙⊗d̄_{n})^{˙⊗m}) ˙⊗(f ˙⊗f̄)^{˙⊗m})∥ ,

and hence by (3.3)

≤ ∥𝔼(σ_{m} ˙⊗σ̄_{m})∥^{1/2}∥𝔼((f ˙⊗f̄)^{˙⊗2m})∥^{1/2}

where we have set

σ_{m} = ∑_{1}^{∞} 𝔼_{n−1}((d_{n} ˙⊗d̄_{n})^{˙⊗m}).

Note that by Lemma 4.2

∥σ_{m}∥_{(2)} ≤ 2∥ ∑_{1}^{∞}(d_{n} ˙⊗d̄_{n})^{˙⊗m}∥_{(2)}

but obviously (recalling (2.4)) ∑(d_{n} ˙⊗d̄_{n})^{˙⊗m} ≺ (∑ d_{n} ˙⊗d̄_{n})^{˙⊗m} and hence (recalling (2.15))

(∑(d_{n} ˙⊗d̄_{n})^{˙⊗m})^{˙⊗2} ≺ S^{˙⊗2m}  (4.9)

so we obtain

∥𝔼(σ_{m} ˙⊗σ̄_{m})∥^{1/2} ≤ 2∥S∥_{(2m)}^{m}.

Thus we find

I(m, 1) ≤ 2∥S∥_{(2m)}^{m}∥f∥_{(4m)}^{2m},

so that (4.8) holds for p = 1 and any m ≥ 1 with C(m, 1) = 2. Let us now denote by (4.8)_{p} the inequality (4.8) meant for a given fixed p but for any m ≥ 1. We will show that for any p ≥ 2

(4.8)_{p/2} ⇒ (4.8)_{p}.

Assuming that m ≥ 1 is fixed, let x_{n} = (d_{n} ˙⊗d̄_{n})^{˙⊗m} and y_{n} = (f_{n−1} ˙⊗f̄_{n−1})^{˙⊗m}. We write

∑ x_{n} ˙⊗y_{n} = a + b
with  a = ∑ 𝔼_{n−1}(x_{n}) ˙⊗y_{n} and b = ∑(x_{n} − 𝔼_{n−1}(x_{n})) ˙⊗y_{n}.

We have I(m, p) = ∥ ∑ x_{n} ˙⊗y_{n}∥_{(p)} and hence

I(m, p) ≤ ∥a∥_{(p)} + ∥b∥_{(p)},  (4.10)

so it suffices to majorize a and b separately. We have by (2.17)

a ≺ ∑ 𝔼_{n−1}(x_{n}) ˙⊗𝔼_{n−1}((f ˙⊗f̄)^{˙⊗m}) = ∑ 𝔼_{n−1}(𝔼_{n−1}(x_{n}) ˙⊗(f ˙⊗f̄)^{˙⊗m}))

and hence by (4.5) and (3.3)

∥a∥_{(p)} ≤ p ∥∑ 𝔼_{n−1}(x_{n}) ˙⊗(f ˙⊗f̄)^{˙⊗m}∥_{(p)}

≤ p∥σ_{m} ˙⊗(f ˙⊗f̄)^{˙⊗m}∥_{(p)}

= p∥σ_{m}^{˙⊗p} ˙⊗(f ˙⊗f̄)^{˙⊗mp}∥_{(1)}^{1/p}

≤ p∥σ_{m}^{˙⊗p}∥(2)^{\frac{1}{p}}∥(f ˙⊗f̄)^{˙⊗mp}∥(2)^{\frac{1}{p}}

= p∥σ_{m}∥_{(2p)}∥f∥_{(4mp)}^{2m}.

But now by (4.5) again

∥σ_{m}∥_{(2p)} ≤ 2p ∥∑(d_{n} ˙⊗d̄_{n})^{˙⊗m}∥_{(2p)}

and hence by (4.9)

∥σ_{m}∥_{(2p)} ≤ 2p∥S∥_{(2mp)}^{m}.

Thus we obtain

∥a∥_{(p)} ≤ p(2p)∥S∥_{(2mp)}^{m}∥f∥_{(4mp)}^{2m}.  (4.11)

We now turn to b. Note that since y_{n} is “predictable” {(x_{n} − 𝔼_{n−1}(x_{n})) ˙⊗y_{n}} is a martingale difference sequence. We will apply the right hand side of (4.1) to b. Note that

S(b) ≈ ∑(x_{n} − 𝔼_{n−1}(x_{n})) ˙⊗\overline{(x_{n} − 𝔼_{n−1}(x_{n}))} ˙⊗y_{n} ˙⊗ȳ_{n},

and hence by (2.7)

\frac{1}{2}S(b) ≺ ∑ x_{n} ˙⊗\overline{x}_{n} ˙⊗y_{n} ˙⊗\overline{y}_{n} + ∑ 𝔼_{n−1}(x_{n}) ˙⊗𝔼_{n−1}(\overline{x}_{n}) ˙⊗y_{n} ˙⊗\overline{y}_{n}.

By (2.10) we get (since y_{n} is predictable)

\frac{1}{2}S(b) ≺ ∑ x_{n} ˙⊗\overline{x}_{n} ˙⊗y_{n} ˙⊗\overline{y}_{n} + ∑ 𝔼_{n−1}(x_{n} ˙⊗\overline{x}_{n} ˙⊗y_{n} ˙⊗\overline{y}_{n})

and hence

\frac{1}{2}∥S(b)∥_{(p/2)} ≤ ∥∑ x_{n} ˙⊗\overline{x}_{n} ˙⊗y_{n} ˙⊗\overline{y}_{n}∥_{(p/2)} + ∥∑ 𝔼_{n−1}(x_{n} ˙⊗\overline{x}_{n} ˙⊗y_{n} ˙⊗\overline{y}_{n})∥_{(p/2)} .

By (4.5) this yields

∥S(b)∥_{(p/2)} ≤ 2(1 + (p/2)) ∥∑ x_{n} ˙⊗\overline{x}_{n} ˙⊗y_{n} ˙⊗\overline{y}_{n}∥_{(p/2)} .

But since

∑ x_{n} ˙⊗\overline{x}_{n} ˙⊗y_{n} ˙⊗\overline{y}_{n} ≈ ∑(d_{n} ˙⊗d̄_{n})^{˙⊗2m} ˙⊗(f_{n−1} ˙⊗f̄_{n−1})^{˙⊗2m}

we may use the induction hypothesis (4.8)_{p/2} (with m replaced by 2m) and we obtain

∥S(b)∥_{(p/2)} ≤ 2(1 + (p/2))C(2m, p/2)∥S∥_{(2mp)}^{2m}∥f∥_{(4mp)}^{4m}.

By the right hand side of (4.1)_{p} we then find

∥b∥_{(p)} ≤ C_{2}(p)∥S(b)∥_{(p/2)}^{1/2}
≤ C′(m, p)∥S∥_{(2mp)}^{m}∥f∥_{(4mp)}^{2m}

for some constant C′(m, p). Thus we conclude by (4.10) and (4.11)

I(m, p) ≤ (p(2p) + C′(m, p))∥S∥_{(2mp)}^{m}∥f∥_{4mp}^{2m}.

In other words we obtain (4.8)_{p}. This completes the proof of (4.8)_{p} for p = 2^{k} by induction on k.

Proof of Theorem 4.1 (General case). We will show that (4.1)_{p} ⇒ (4.1)_{2p}. We again start from

f ˙⊗f̄− S = a + b

where a = ∑ d_{n} ˙⊗f̄_{n−1} and b = ∑ f_{n−1} ˙⊗d̄_{n}. By the right hand side of (4.1)_{p} we have

∥f ˙⊗f̄− S∥_{(p)} ≤ 2C_{2}(p) ∥∑ d_{n} ˙⊗d̄_{n} ˙⊗f_{n−1} ˙⊗f̄_{n−1}∥_{(p/2)}^{1/2}

and hence by (4.8)

≤ 2C_{2}(p)C(1, p/2)^{1/2}∥S∥_{(p)}^{1/2}∥f∥_{(2p)}.

Thus we find a fortiori setting C′′ = 2C_{2}(p)C(1, p/2)^{1/2}

|∥f ˙⊗f̄∥_{(p)} − ∥S∥_{(p)}| ≤ C′′∥S∥_{(p)}^{1/2}∥f∥_{(2p)}.

Thus setting again x = ∥f∥_{(2p)}, y = ∥S∥_{(p)}^{1/2} we find

|x^{2} − y^{2}| ≤ C′′xy

and we conclude that x and y must be equivalent quantities, or equivalently that (4.1)_{2p} holds. By induction this completes the proof.

5 Burkholder-Rosenthal inequality

Let 2 < p < ∞ be fixed. The usual form of the Burkholder-Rosenthal inequality expresses the equivalence, for scalar valued martingales, of ∥ ∑ d_{n}∥_{p} and

BR_{∞} = ∥σ∥_{p} + ∥ sup |d_{n}|∥_{p}.  (5.1)

It is easy to deduce from that the equivalence of that same norm with

BR_{q} = ∥σ∥_{p} + ∥(∑ |d_{n}|^{q})^{1/q}∥_{p}  (5.2)

for any q such that 2 < q ≤ ∞. Indeed, we have obviously BR_{∞} ≤ BR_{q}. Conversely, using (here \frac{1}{q} = \frac{1−θ}{2} + \frac{θ}{∞})

∥(∑ |d_{n}|^{q})^{1/q}∥_{p} ≤ ∥S∥_{p}^{1−θ}∥ sup |d_{n}|∥_{p}^{θ}

and the equivalence ∥S∥_{p} ≃ ∥ ∑ d_{n}∥_{p}, one can easily deduce that there is a constant C′ such that

BR_{q} ≤ C′∥ ∑ d_{n}∥_{p}^{1−θ}BR_{∞}^{θ}.

Thus an inequality of the form

∥∑ d_{n}∥_{p} ≤ CBR_{∞}

implies “automatically”

BR_{∞} ≤ BR_{q} ≤ C′C^{1−θ}BR_{∞}.

Similarly,

∥∑ d_{n}∥_{p} ≤ CBR_{q} ⇒ ∥∑ d_{n}∥_{p} ≤ CC′∥ ∑ d_{n}∥_{p}^{1−θ}BR_{∞}^{θ} ⇒

⇒ ∥∑ d_{n}∥_{p} ≤ (CC′)^{1/θ}BR_{∞}.

Thus, modulo simple manipulations, the Burkholder-Rosenthal inequality reduces to the equivalence for some q such that 2 < q ≤ ∞ of ∥ ∑ d_{n}∥_{p} and BR_{q}. Note that the one sided inequality expressing that BR_{q} = ∥σ∥_{p} + ∥(∑ |d_{n}|^{q})^{1/q}∥_{p} is dominated by ∥ ∑ d_{n}∥_{p} reduces obviously to

∥σ∥_{p} ≤ C ∥∑ |d_{n}|^{2}∥_{p/2}^{1/2}

that holds for p ≥ 2 by Burkholder–Davis–Gundy dualization of Doob’s inequality. Therefore, the novelty of the Burkholder-Rosenthal inequality is the fact that there is a constant C′′′ such that

∥ ∑ d_{n}∥_{p} ≤ C′′′BR_{q}.

In the original Rosenthal inequality, restricted to sums of independent d_{n}’s, or in the non-commutative version of [17, 19], the value q = p is the most interesting choice. In the inequalities below, for p = 2^{k} ≥ 4, we will work with q = 4.

We will use the following extension of (3.3).

Proposition 5.1. For any integer m ≥ 1 and finite sequences (a_{k}), (b_{k}) in B(H) ⊗ L_{2m} we have

∥𝔼 ((∑ a_{k} ˙⊗b_{k})^{˙⊗m})∥ ≤ ∥𝔼 ((∑ a_{k} ˙⊗ā_{k})^{˙⊗m})∥^{1/2} ∥𝔼 ((∑ b_{k} ˙⊗b̄_{k})^{˙⊗m})∥^{1/2} .

(5.3)

More generally, consider finite sequences (a_{k}^{(j)}), (b_{k}^{(j)}) in B(H) ⊗ L_{2m} for j = 1, …, m. Let T_{j} = ∑_{k} a_{k}^{(j)} ˙⊗ b_{k}^{(j)} and let α_{j} = ∑_{k} a_{k}^{(j)} ⊗ \overline{a_{k}^{(j)}} and β_{j} = ∑_{k} b_{k}^{(j)} ⊗ \overline{b_{k}^{(j)}}. We have then

∥𝔼(T_{1} ˙⊗ ⋯ ˙⊗ T_{m})∥ ≤ ∥𝔼(α_{1} ˙⊗ ⋯ ˙⊗ α_{m})∥^{1/2}∥𝔼(β_{1} ⊗ ⋯ ⊗ β_{m})∥^{1/2}. (5.4)

Proof. Up to permutation, 𝔼((∑ a_{k} ˙⊗b_{k})^{˙⊗m}) is the same as

𝔼 ( ∑_{k(1),...,k(m)} a_{k(1)} ˙⊗ ⋯ ˙⊗a_{k(m)} ˙⊗b_{k(1)} ˙⊗ ⋯ ˙⊗b_{k(m)}) .

Therefore, by (3.3) we have

∥𝔼 ((∑ a_{k} ˙⊗b_{k})^{˙⊗m})∥ ≤ ∥𝔼 ∑_{k(1),...,k(m)} a_{k(1)} … a_{k(m)} ˙⊗ā_{k(1)} … ā_{k(m)}∥^{\frac{1}{2}}

× ∥𝔼 ∑_{k(1),...,k(m)} b_{k(1)} … b_{k(m)} ˙⊗b̄_{k(1)} … b̄_{k(m)}∥^{\frac{1}{2}}

= ∥𝔼 ((∑ a_{k} ˙⊗ā_{k})^{˙⊗m})∥^{\frac{1}{2}} ∥𝔼 ((∑ b_{k} ˙⊗b̄_{k})^{˙⊗m})∥^{\frac{1}{2}} .

Up to permutation T_{1} ˙⊗ ⋯ ˙⊗ T_{m} is the same as

∑_{k(1),...,k(m)} a_{k(1)}^{(1)} ˙⊗ ⋯ ˙⊗ a_{k(m)}^{(m)} ˙⊗ b_{k(1)}^{(1)} ˙⊗ ⋯ ˙⊗ b_{k(m)}^{(m)},

which can be written as ∑_{k} a_{k} ˙⊗ b_{k} with k = (k(1), …, k(m)), a_{k} = a_{k(1)}^{(1)} ˙⊗ ⋯ ˙⊗ a_{k(m)}^{(m)} and b_{k} = b_{k(1)}^{(1)} ˙⊗ ⋯ ˙⊗ b_{k(m)}^{(m)}. Therefore (5.4) follows from the m = 1 case of (5.3).

Remark 5.2. Let H = ℓ_{2} and B = B(H). The preceding Proposition shows that

∥∑ a_{k} ˙⊗ā_{k}∥(m)^{\frac{1}{2}} = sup \{∥∑ a_{k} ˙⊗b_{k}∥_{(m)}\}  (5.5)

where the supremum runs over the set D of all finite sequences (b_{k}) in B ⊗ L_{2m} such that ∥ ∑ b_{k} ˙⊗b_{k}∥_{(m)} ≤ 1. (Indeed the sup is attained for b_{k} = ā_{k}, suitably normalized.) Thus (5.5) allows us to define an o.s.s. on the space L_{2m}(Ω, μ; ℓ_{2}), corresponding to “Λ_{2m} with values in OH”. Indeed, we can proceed as before for Λ_{p}: we consider the subspace E_{0} ⊂ L_{2m} ⊗ ℓ_{2} formed of all finite sums ∑ a_{k} ⊗ e_{k} (a_{k} ∈ L_{2m}) and we define

J: E_{0} ⟶ ⨁_{(b_{k})∈D}(Λ_{m} ⊗_{min} B)

by

J (∑ a_{k} ⊗ e_{k}) = ⨁_{(b_{k})∈D} ∑ a_{k} ˙⊗b_{k}.

This produces an o.s.s. on L_{2m}(μ; ℓ_{2}). It is easy to see that if a ∈ L_{2m} is fixed in the unit sphere, the restriction of J to a ⊗ ℓ_{2} induces on ℓ_{2} the o.s.s. of OH while if x ∈ ℓ_{2} is fixed in the unit sphere, restricting J to L_{2m} ⊗ x induces on L_{2m} the o.s.s. of Λ_{2m}. Note in passing that, in sharp contrast with [23], except for the preceding special case, we do not have any reasonable definition to propose for the “vector valued” analogue of the Λ_{p} spaces.

As a consequence we find an analogue of Stein’s inequality (here we could obviously replace 𝔼_{n−1} by 𝔼_{n}):

Corollary 5.3. Let x_{n} be an arbitrary finite sequence in B(H) ⊗ L_{4m}. Let v = ∑ 𝔼_{n−1}(x_{n} ˙⊗x̄_{n}) ˙⊗𝔼_{n−1}(x̄_{n} ˙⊗x_{n}) and δ = ∑ x_{n} ˙⊗x̄_{n} ˙⊗x̄_{n} ˙⊗x_{n}. Then for any integer m ≥ 1

∥𝔼(v^{˙⊗m})∥ ≤ m^{m}∥𝔼(δ^{˙⊗m})∥.  (5.6)

Proof. Let w = ∑ 𝔼_{n−1}(x_{n} ˙⊗x̄_{n} ˙⊗x̄_{n} ˙⊗x_{n}). By (2.10) we have v ≺ w. Then by (2.15), (2.11), Lemma 2.3 and (4.5), we have

∥𝔼(v^{˙⊗m})∥ ≤ ∥𝔼(w^{˙⊗m})∥ ≤ m^{m}∥𝔼(δ^{˙⊗m})∥.

Lemma 5.4. Let p = 2^{k} ≥ 4 as before. Let δ = ∑ d_{n} ˙⊗d̄_{n} ˙⊗d̄_{n} ˙⊗d_{n}. There is a constant C_{4}(p) such that

∥𝔼(S^{˙⊗p/2})∥^{1/p} ≤ C_{4}(p)[∥𝔼(σ^{˙⊗p/2})∥^{\frac{1}{p}} + ∥𝔼(δ^{˙⊗p/4})∥^{\frac{1}{p}}].  (5.7)

Proof. Note that

S − σ = ∑ dc_{n}

where c_{n} = d_{n} ˙⊗d̄_{n} − 𝔼_{n−1}(d_{n} ˙⊗d̄_{n}). Thus by the right hand side of (4.1) we have

∥S − σ∥_{B⊗_{min}\overline{B}⊗_{min}Λ_{p/2}} ≤ C_{2}(p/2)∥S(c)∥_{B⊗_{min}\overline{B}⊗_{min}\overline{B}⊗_{min}B⊗_{min}Λ_{p/4}}^{1/2}.

By (2.7)

\frac{1}{2}dc_{n} ˙⊗dc̄_{n} ≺ d_{n} ˙⊗d̄_{n} ˙⊗d̄_{n} ˙⊗d_{n} + 𝔼_{n−1}(d_{n} ˙⊗d̄_{n}) ˙⊗𝔼_{n−1}(d̄_{n} ˙⊗d_{n})

therefore

\frac{1}{2}S(c) ≺ δ + v,

where we now set v = ∑ 𝔼_{n−1}(d_{n} ˙⊗d̄_{n}) ˙⊗𝔼_{n−1}(d̄_{n} ˙⊗d_{n}). Thus we find

|∥S∥_{B⊗_{min}\overline{B}⊗_{min}Λ_{p/2}} − ∥σ∥_{B⊗_{min}\overline{B}⊗_{min}Λ_{p/2}} | ≤ ||S − σ∥_{B⊗_{min}\overline{B}⊗_{min}Λ_{p/2}}

≤ C_{2}(p/2)\sqrt{2}(∥δ∥•^{\frac{1}{2}} + ∥v∥•^{\frac{1}{2}} )

where ∥ ∥_{•} is the norm in B ⊗_{min} \overline{B} ⊗_{min} \overline{B} ⊗_{min} B ⊗_{min} Λ_{p/4}. By (5.6) we have

∥v∥_{•} ≤ (p/4)∥δ∥_{•}

and hence

∥S∥_{B⊗\overline{B}⊗Λ_{p/2}} ≤ ∥σ∥_{B⊗\overline{B}⊗Λ_{p/2}} + C_{2}(p/2)\sqrt{2}(1 + (p/4)^{\frac{1}{2}} )∥δ∥•^{\frac{1}{2}} .

Taking the square root of the last inequality we obtain (5.7).

We now give a version (corresponding to BR_{q} with q = 4) for Λ_{p} of the Burkholder-Rosenthal inequality :

Theorem 5.5. For any p ≥ 4 of the form p = 2^{k} for some k ≥ 1 there are positive constants C′_{1}(p), and C′_{2}(p) such that for any test function f in B ⊗ L_{p} we have

C′_{1}(p)^{−1}[f]_{p} ≤ ∥f∥_{B⊗_{min}Λ_{p}} ≤ C′_{2}(p)[f]_{p}  (5.8)

where

[f]_{p} = ∥σ(f)∥_{B⊗_{min}\overline{B}⊗_{min}Λ_{p/2}}^{1/2} + ∥𝔼(∑ d_{n} ˙⊗d̄_{n} ˙⊗d_{n} ˙⊗d̄_{n})^{˙⊗p/4}∥^{1/p} .

Proof. Note that \overline{S} and S are the same after a transposition of the two factors, thus the same is true for S ˙⊗\overline{S} and S^{˙⊗2}, and we have

∥𝔼(S ˙⊗\overline{S})∥ = ∥𝔼(S^{˙⊗2})∥

and similarly for any even m ≥ 1

𝔼((S ˙⊗\overline{S})^{˙⊗m/2})∥ = ∥𝔼(S^{˙⊗m})∥.

Recall that in a suitable permutation we may write 0 ≺ d_{n} ˙⊗d̄_{n} ˙⊗d̄_{k} ˙⊗d_{k} for all n, k and hence

∑ d_{n} ˙⊗d̄_{n} ˙⊗d̄_{n} ˙⊗d_{n} ≺ ∑_{n,k} d_{n} ˙⊗d̄_{n} ˙⊗d̄_{k} ˙⊗d_{k} = S ˙⊗\overline{S},

and hence for any even integer m

(∑ d_{n} ˙⊗d̄_{n} ˙⊗d̄_{n} ˙⊗d_{n})^{˙⊗m/2} ≺ S^{˙⊗m}.

Therefore

∥𝔼 (∑ d_{n} ˙⊗d̄_{n} ˙⊗d_{n} ˙⊗d̄_{n})^{˙⊗m/2}∥ ≤ ∥𝔼(S^{˙⊗m})∥.  (5.9)

Let σ = σ(f). Now if p = 2m, (4.5) implies

∥σ∥_{B⊗_{min}\overline{B}⊗_{min}Λ_{p/2}}^{1/2} = ∥𝔼(σ^{˙⊗m})∥^{\frac{1}{2m}} ≤ m^{1/2}∥𝔼(S^{˙⊗m})∥^{\frac{1}{2m}} ,  (5.10)

and hence by (5.9) and (5.10)

[f]_{p} ≤ (m^{1/2} + 1)∥𝔼(S^{˙⊗m})∥^{1/2m}
= (m^{1/2} + 1)∥S∥_{B⊗_{min}\overline{B}⊗_{min}Λ_{m}}^{1/2}.

Thus the left hand side of (5.8) follows from (4.1). Since the converse inequality follows from Lemma 5.4 and (4.1), this completes the proof.

6 Hilbert transform

Consider the Hilbert transform on L_{p}(𝕋, dm). We will show that this defines a completely bounded operator on Λ_{p}(𝕋, m) again for p ≥ 2 of the form p = 2^{k} with k ∈ ℕ. The proof is modeled on Marcel Riesz’s proof as presented in Zygmund’s classical treatise on trigonometric series. One of the first references using this trick in a broader context is Cotlar’s paper [10]. Let f be a trigonometric polynomial with coefficients in B(H), i.e. f = ∑_{n∈ℤ} f̂(n)e^{int} with f̂: ℤ → B(H) finitely supported. The Hilbert transform T f is defined by

T f = ∑_{n∈ℤ} φ(n)f̂(n)e^{int}  (6.1)

where φ(0) = 0 and

φ(n) = −i sign(n).  ∀n ∈ ℤ

Note that T^{2} = −id on the subspace {f | f̂(0) = 0}. We will use the following classical identity valid for any pair f, g of complex valued trigonometric polynomials

T (fg − (T f)(T g)) = fT g + (T f)g.  (6.2)

This can be checked easily as a property of φ since it reduces to the case f = z^{n}, g = z^{m} (n, m ∈ ℤ). A less pedestrian approach is to recall that if f is real valued, T f is characterized as the unique real valued v, the “conjugate function”, actually here also a trigonometric polynomial, such that v̂(0) = 0 and z ↦ f(z) + iv(z) is the boundary value of an analytic function (actually a polynomial in z) inside the unit disc D. Then (6.2) boils down to the observation that since (f + iT f)(g + iT g) is the product of two analytic functions on D, f(T g) + (T f)g must be the “conjugate” of fg − (T f)(T g). The complex case follows from the real one: for a complex valued f, we define T f = T (ℜ(f)) + iT (ℑ(f)) and (6.2) remains valid. From (6.2) in the ℂ-valued case, it is immediate to deduce that for any pair f, g of B(H)-valued trigonometric polynomials we have

T (f ˙⊗g − (T f) ˙⊗(T g)) = f ˙⊗(T g) + (T f) ˙⊗g  (6.3)

where (as before) the notation f ˙⊗g stands for the B(H)⊗B(H) valued function z → f(z)⊗g(z) on 𝕋, and where we still denote by T the mapping (that should be denoted by T ⊗ I) taking f ⊗ b (f ∈ L_{2}, b ∈ B(H)) to (T f) ⊗ b. Now, it is a simple exercise to check that for any such f

\overline{T f} = T (f̄)

(this is an equality between two \overline{B(H)} valued functions). Therefore we have also:

T (f ⊗ ḡ − (T f) ˙⊗T (ḡ)) = f ⊗ T (ḡ) + T f ⊗ ḡ.  (6.4)

We can now apply the well known Riesz–Cotlar trick to our situation:

Theorem 6.1. For any p ≥ 2, of the form p = 2^{k} with k ∈ ℕ, the Hilbert transform T is a c.b. mapping on Λ_{p}(𝕋, m).

Proof. If we restrict (as we may) to functions such that f̂(0) = 0, we have T^{2} = −id and hence (6.4) implies

T f ˙⊗T f̄− f ˙⊗f̄ = T (f ˙⊗(\overline{T f}) + (T f) ˙⊗f̄).  (6.5)

We can then again use induction on k. Assume the result known for p, i.e. that there is a constant C such that

∥T f∥_{B(H)⊗_{min}Λ_{p}} ≤ C∥f∥_{B(H)⊗_{min}Λ_{p}}.

We will prove that the same holds for 2p in place of p (with a different constant). Let B = B(H). By (6.5), we have

∥T f ˙⊗\overline{T f} − f ⊗ f̄∥_{B⊗_{min}\overline{B}⊗_{min}Λ_{p}} ≤ 2C∥f ⊗ \overline{T f}∥_{B⊗_{min}\overline{B}⊗Λ_{p}}.

By (3.5), this term is

≤ 2C∥f ⊗ f̄∥_{B⊗_{min}\overline{B}⊗_{min}Λ_{p}}^{1/2}∥T f ⊗ \overline{T f}∥_{B⊗_{min}\overline{B}⊗Λ_{p}}^{1/2}.

We have

∥f∥_{B⊗_{min}Λ_{2p}}^{2} = ∥f ˙⊗\overline{f}∥_{B⊗_{min}\overline{B}⊗_{min}Λ_{p}}

and

∥T f∥_{B⊗_{min}Λ_{2p}}^{2} = ∥T f ˙⊗\overline{T f}∥_{B⊗_{min}\overline{B}⊗_{min}Λ_{p}}.

Therefore, denoting this time x = ∥T f∥_{B⊗_{min}Λ_{2p}} and y = ∥f∥_{B⊗_{min}Λ_{2p}}, and using

| ∥T f ˙⊗\overline{T f}∥_{(p)} − ∥f ⊗ f̄∥_{(p)}| ≤ ∥T f ˙⊗\overline{T f} − f ⊗ f̄∥_{(p)}

we find again

|x^{2} − y^{2}| ≤ 2Cxy.

Thus we conclude that x and y are “equivalent,” completing the proof with 2p in place of p.

7 Comparison with L_{p} L_{p} L_{p}

Let B = B(H) with (say) H = ℓ_{2}. Let E_{1}, ⋯, E_{m} and G be operator spaces. Recall that an m-linear mapping

u : E_{1} × ⋯ × E_{m} → G

is called (jointly) completely bounded (j.c.b. in short) if the associated m-linear mapping from

û: (B ⊗_{min} E_{1}) × ⋯ × (B ⊗_{min} E_{m}) → B ⊗_{min} ⋯ ⊗_{min} B ⊗_{min} G

is bounded. We set ∥u∥_{cb} = ∥û∥, and we say that u is (jointly) completely contractive if ∥u∥_{cb} ≤ 1. Note the obvious stability of these maps under composition: for instance if F, L are operator spaces and if v: G × F → L is bilinear and j.c.b. then the (m + 1)-linear mapping w; E_{1} × ⋯ × E_{m} × F → L defined by

w(x_{1}, ⋯, x_{m}, y) = v(u(x_{1}, ⋯, x_{m}), y)

is also j.c.b. with ∥w∥_{cb} ≤ ∥u∥_{cb}∥v∥_{cb}. Moreover, if in the above definition we replace B by the space K of compact operators on ℓ_{2}, the definition and the value of ∥u∥_{cb} is unchanged. This allows to extend (following [23]) the complex interpolation theorem for multilinear mappings. In particular, we have Lemma 7.1. Let 1 ≤ p, q, r ≤ ∞ be such that 1/p + 1/q = 1/r. Then the pointwise product from L_{p} ×L_{q} to L_{r} is completely contractive. More generally, if 1 ≤ p_{j} ≤ ∞ (1 ≤ j ≤ N) are such that ∑ 1/p_{j} = 1/r, the product map L_{p_{1}} × ⋯ × L_{p_{N}} → L_{r} is completely contractive. In particular, if p is any positive integer, the pointwise product P_{p} from L_{p} × ⋯ × L_{p} (p-times) to L_{1} is completely contractive.

Proof. The three cases either q = ∞, p = r or p = ∞, q = r or q = p′, r = 1 are obvious. By interpolation and then exchanging the roles of p and q, this implies the general case. By the preceding remark, one can iterate and the second assertion becomes clear.

Theorem 7.2. Let p = 2m, m ∈ ℕ. The identity map L_{p} → Λ_{p} is completely contractive.

Proof. By the preceding Lemma 7.1, P_{p}: L_{p} × ⋯ × L_{p} → L_{1} is completely contractive. Therefore

∥∫ f_{1} ˙⊗ ⋯ ˙⊗f_{p/2} ˙⊗f̄_{1} ˙⊗ ⋯ ˙⊗f̄_{p/2}∥ ≤ (∏_{1}^{p/2} ∥f_{j}∥_{B⊗_{min}L_{p}} )^{2} .

Thus taking f_{1} = ⋯ = f_{p/2} we get by (3.8)

∥f∥_{B⊗_{min}Λ_{p}}^{p} ≤ ∥f∥_{B⊗_{min}L_{p}}^{p}

and we obtain ∥L_{p} → Λ_{p}∥_{cb} = 1.

Remark 7.3. The preceding argument (together with Corollary 3.6) shows that the o.s.s. on Λ_{p} is essentially the minimal one on L_{p} such that P_{p}: Λ_{p} × ⋯ × Λ_{p} → L_{1} is completely contractive. More precisely, assume p ∈ ℕ. Let Q_{p}: L_{p} × ⋯ × L_{p} × L̄_{p} × ⋯ × L̄_{p} (where L_{p} and L̄_{p} are repeated p/2 times) be the p-linear mapping taking (f_{1}, ⋯, f_{p/2}, ḡ_{1}, ⋯, ḡ_{p/2}) to ∫ f_{1} ⋯ f_{p/2}ḡ_{1}, ⋯, ḡ_{p/2}dμ. Then if X_{p} is an o.s. isometric to L_{p}, such that Q_{p}: X_{p} × ⋯ × X_{p} × X̄_{p} × ⋯ × X̄_{p} → L_{1} is completely contractive, the identity map X_{p} → Λ_{p} is completely contractive. In the case of L_{p} itself with its interpolated o.s.s. we could consider P_{p} instead of Q_{p} because the map f ↦ f̄ is a completely isometric antilinear isomorphism from L_{p} to itself, and hence defines a completely isometric linear isomorphism from L_{p} to L̄_{p}. (This can be checked easily by interpolation starting from p = ∞ and-by duality-p = 1.)

This remark leads to:

Corollary 7.4. For any integer p ≥ 1, we have a completely contractive inclusion

(Λ_{p}, L_{∞})_{1/2} → Λ_{2p}.

Proof. Let X_{2p} = (Λ_{p}, L_{∞})_{1/2}. We will argue as in the proof of Lemma 7.1, applying complex interpolation to the product map P_{2p} : (x_{1}, ⋯, x_{2p}) ↦ x_{1} ⋯ x_{2p}. Clearly, by Lemma 7.1, P_{2p} is completely contractive both as a map from (Λ_{p})^{p} × (L̄_{∞})^{p} to L_{1} and as one from (L_{∞})^{p} × (Λ̄_{p})^{p} to L_{1}. Therefore, by interpolation, P_{2p}: X_{2p} × ⋯ × X_{2p} × X̄_{2p} × ⋯ × X̄_{2p} → L_{1} is completely contractive. The preceding remark (applied with 2p in place of p) then yields this corollary.

Remark 7.5. We wish to compare here the operator spaces L_{p} and Λ_{p}. We already know that they are different since the Khintchine inequalities lead to two different operator spaces in both cases, but we can give a more precise quantitative estimate. Let us denote by L_{p}^{n} the space L_{p}(Ω_{n}, μ_{n}) when Ω_{n} = [1, …, n] and μ_{n} is the uniform probability measure on Ω. We then set

Λ_{p}^{n} = Λ_{p}(Ω_{n}, μ_{n}).

We claim that for any even integer p > 2, there is δ_{p} > 0 such that for any n the identity map (denoted id) satisfies

∥id: Λ_{p}^{n} → L_{p}^{n}∥_{cb} ≥ δ_{p}n^{\frac{1}{p} ( \frac{1}{2} − \frac{1}{p})}.

To prove this, we will use an adaptation (with Λ_{p} instead of L_{p}) of the results in [13, 24]. Indeed, by Corollary 14.2 below, using the classical “Rudin examples” of Λ(p)-sets, one can show that the space Λ_{p}^{n} contains a subspace E_{n} ⊂ Λ_{p}^{n} with dim E_{n} = d(n) ≥ n^{2/p} and such that the inclusion OH_{d(n)} ⊂ E_{n} satisfies ∥OH_{d(n)} → E_{n}∥_{cb} ≤ χ_{p}. Moreover, there is a projection P_{n}: Λ_{p}^{n} → E_{n} with ∥P_{n}∥_{cb} ≤ χ_{p}. Here p is an even integer > 2 and χ_{p} is a constant depending only on p. In addition, by [13], the same space E_{n} considered in L_{p}^{n} is (uniformly over n) completely isomorphic to R_{p}(d(n)) ∩ C_{p}(d(n)) (intersection of row and column space in S_{p}^{d(n)}). In fact we use only the easy direction of this result, namely that ∥E_{n} → R_{p}(d(n))∥_{cb} ≤ 1 and ∥E_{n} → C_{p}(d(n))∥_{cb} ≤ 1. It follows that there is a constant δ_{p} > 0 such that if id denotes the identity map we have

∥id: Λ_{p}^{n} → L_{p}^{n}∥_{cb} ≥ δ_{p}∥OH_{d(n)} → C_{p}(d(n))∥_{cb}.

Recall that (see [25, p. 219]) ∥OH(d) → C_{∞}(d)∥_{cb} = d^{1/4}. Thus by interpolation, we have for any d if \frac{1}{p} = \frac{θ}{2}

∥C_{p}(d) → C_{∞}(d)∥_{cb} ≤ ∥OH(d) → C_{∞}(d)∥_{cb}^{θ} = d^{θ/4},

also

∥OH(d) → C_{p}(d)∥_{cb}∥C_{p}(d) → C_{∞}(d)∥_{cb} ≥ ∥OH(d) → C_{∞}(d)∥_{cb},

therefore we find

∥OH(d) → C_{p}(d)∥_{cb} ≥ d^{1/4}d^{−θ/4}

and we conclude for some δ′_{p} > 0

∥id: Λ_{p}^{n} → L_{p}^{n}∥_{cb} ≥ δ′_{p}(n^{2/p})^{\frac{1−θ}{4}} = δ′_{p}n^{\frac{θ(1−θ)}{4}} .

A similar argument applies to compare Λ_{p}^{n} with either min(L_{p}^{n}) or max(L_{p}^{n}). Using the projections P_{n}, we easily deduce that for some constant χ′_{p} > 0

∥Λ_{p}^{n} → max(L_{p}^{n})∥_{cb} ≥ χ′_{p}∥OH_{d(n)} → max(ℓ_{2}^{d(n)})∥_{cb}

and

∥ min(L_{p}^{n}) → Λ_{p}^{n}∥_{cb} ≥ χ′_{p}∥ min(ℓ_{2}^{d(n)}) → OH_{d(n)}∥_{cb}.

But it is known (see [25, p. 220]) that for any d

∥OH_{d} → max(ℓ_{2}^{d})∥_{cb} = ∥ min(ℓ_{2}^{d}) → OH_{d}∥_{cb} ≃ cd^{1/2}

where c > 0 is independent of d. Thus we obtain

∥Λ_{p}^{n} → max(L_{p}^{n})∥_{cb} ≥ cχ′_{p}d(n)^{1/2} ≃ c′n^{1/p}

and similarly

∥ min(L_{p}^{n}) → Λ_{p}^{n}∥_{cb} ≥ c′n^{1/p}.

8 The non-commutative case

Let M be a von Neumann algebra equipped with a normal semi-finite faithful trace τ, and let L_{p}(τ) be the associated “non-commutative” L_{p}-space. The preceding procedure works equally well in the non-commutative case, but requires a little more care. To define the o.s.s. on L_{p}(τ) that will be of interest to us we consider f in B(H) ⊗ L_{p}(τ) of the form f = ∑_{1}^{n} b_{k} ⊗ x_{k} and we define f^{∗} ∈ \overline{B(H)} ⊗ L_{p}(τ) by

f^{∗} = ∑_{1}^{n} b̄_{k} ⊗ x_{k}^{∗}.

Consider f = ∑_{1}^{n} b_{k} ⊗x_{k} ∈ B(H)⊗L_{p}(τ) as above and g = ∑ c_{j} ⊗y_{j} ∈ B(K)⊗ L_{q}(τ) (p, q ≥ 1). We denote by f ˙⊗g ∈ B(H)⊗B(K)⊗L_{r}(τ) (r ≥ 1, \frac{1}{r} = \frac{1}{p} + \frac{1}{q}) the element defined by

f ˙⊗g = ∑_{k,j} b_{k} ⊗ c_{j} ⊗ x_{k}y_{j}.

Given f ∈ B(H) ⊗ L_{1}(τ) we denote τ̂ = id_{B(H)} ⊗ τ: B(H) ⊗ L_{1}(τ) → B(H). More explicitly if f is as above (here p = 1) we set

τ̂(f) = ∑ b_{k}τ(x_{k}),

and since the norm and the cb-norm coincide for linear forms, we have

∥τ̂(f)∥ ≤ ∥f∥_{B⊗_{min}L_{1}(τ)}.

Then, by the trace property, if r = 1, τ̂(f ˙⊗g) and τ̂(g ˙⊗f) are the same up to transposition of the two factors, and hence have the same minimal norm. More generally, given finite sequences f_{ℓ} ∈ B(H) ⊗ L_{p}(τ) as above and g_{ℓ} ∈ B(K) ⊗ L_{q}(τ) (r ≥ 1, 1/r = 1/p + 1/q), the same reasoning yields

∥τ̂(∑_{ℓ} f_{ℓ} ˙⊗g_{ℓ})∥ = ∥τ̂(∑_{ℓ} g_{ℓ} ˙⊗f_{ℓ})∥.  (8.1)

This identity (8.1) will considerably facilitate the generalization of most of the preceding proofs to the non-commutative case, in a rather easier fashion than for the corresponding steps in [26]. Now, Haagerup’s version of the Cauchy–Schwarz inequality for the Hilbert space ℓ_{2}(L_{2}(τ)) becomes:

Lemma 8.1. Let f_{k}, g_{k} ∈ B ⊗ L_{2}(τ) (k = 1, …, N). Then

∥∑_{1}^{N} τ̂(f_{k}^{∗} ˙⊗g_{k})∥ ≤ ∥∑ τ̂(f_{k}^{∗} ˙⊗f_{k})∥^{1/2} ∥∑ τ̂ (∑ g_{k}^{∗} ˙⊗g_{k})∥^{1/2} ,  (8.2)

and actually this is valid when τ is any (not necessarily tracial) state on M.

Proof. Let H be the Hilbert space obtained (after quotient and completion) from M equipped with the scalar product ⟨x, y⟩ = τ(y^{∗}x). Then this lemma appears as a particular case of (2.2).

We will use repeatedly the identification

\overline{B(H)} = B(\overline{H}).

The operator space Λ_{p}(τ) will be defined as isometric to L_{p}(τ) but with an o.s.s. such that for any f in B(H) ⊗ L_{p}(τ) (p an even integer) we have

∥f∥_{B(H)⊗_{min}Λ_{p}(τ)} = ∥τ̂(f^{∗} ˙⊗f ⊗ ⋯ ˙⊗f^{∗} ˙⊗f)∥B(\overline{H}⊗_{2}H⊗_{2}···⊗_{2}\overline{H}⊗_{2}H)^{\frac{1}{p}} (8.3)

where f^{∗} ˙⊗f and \overline{H} ⊗_{2} H are repeated p/2-times. To prove that (8.3) really defines a norm (and an o.s.s.) on B(H) ⊗ L_{p}(τ) we proceed exactly as in the commutative case by first establishing a Hölder type inequality:

Lemma 8.2. Let p ≥ 2 be an even integer. Consider f_{j} ∈ B(H_{j}) ⊗ L_{p}(τ). Let

∥f_{j}∥_{(p)} = ∥τ̂(f_{j}^{∗} ˙⊗f_{j} ˙⊗ ⋯ ˙⊗f_{j}^{∗} ˙⊗f_{j})∥_{B(\overline{H}_{j}⊗H_{j}⊗···⊗\overline{H}_{j}⊗H_{j})}^{1/p}

where f_{j}^{∗} ˙⊗f_{j} is repeated p/2 times. We have then

∥τ̂(f_{1} ˙⊗ ⋯ ˙⊗f_{p})∥ ≤ ∏_{j=1}^{p} ∥f_{j}∥_{(p)}.  (8.4)

Proof. We will use repeatedly the fact that the minimal tensor product is commutative i.e. a permutation σ of the factors induces a complete isometry (and actually a ∗-isomorphism) from B(H_{1} ⊗_{2} ⋯ ⊗ H_{n}) to

B(H_{σ(1)} ⊗_{2} ⋯ ⊗_{2} H_{σ(n)}).

Thus for any x = ∑ b_{j}^{1} ⊗ ⋯ ⊗ b_{j}^{n} ⊗ x_{j} ∈ B(H_{1}) ⊗ ⋯ ⊗ B(H_{n}) ⊗ L_{p}(τ), if we denote σ[x] = ∑ b_{j}^{σ(1)} ⊗ ⋯ ⊗ b_{j}^{σ(n)} ⊗ x_{j} we have

∥x∥_{min} = ∥σ[x]∥_{min}.

Let y = σ[x]. To indicate that one can pass from x to y by a permutation, it will be convenient to write x ≈ y. Thus x ≈ y guarantees ∥x∥_{min} = ∥y∥_{min}. For example, let

f_{1} ∈ B(H_{1}) ⊗ L_{p}(τ) f_{2} ∈ B(H_{2}) ⊗ L_{p}(τ).

Then τ̂((f_{1} ˙⊗f_{2})^{∗}) ≈ τ̂(f_{2}^{∗} ˙⊗f_{1}^{∗}) and hence

∥τ̂((f_{1} ˙⊗f_{2})^{∗})∥ = ∥τ̂(f_{2}^{∗} ˙⊗f_{1}^{∗})∥.

Also using the trace property we have for any f in B(H) ⊗ L_{2}(τ)

τ̂(f^{∗} ˙⊗f) ≈ τ̂(f ˙⊗f^{∗})

and hence

∥f∥_{(2)} = ∥f^{∗}∥_{(2)}.  (8.5)

More generally, for any f_{1}, …, f_{p} as before we have by (8.1)

∥τ̂(f_{1} ˙⊗ ⋯ ˙⊗f_{p})∥_{B(H_{1}⊗_{2}···⊗_{2}H_{p})} = ∥τ̂(f_{p} ˙⊗f_{1} ˙⊗ ⋯ ˙⊗f_{p−1})∥_{B(H_{p}⊗_{2}···⊗_{2}H_{p−1})}.

(8.6)

In particular this gives us for any j

∥f_{j}∥_{(p)} = ∥τ̂(f_{j} ˙⊗f_{j}^{∗} ˙⊗ ⋯ ˙⊗f_{j} ˙⊗f_{j}^{∗})∥^{\frac{1}{p}}  (8.7)

or equivalently

∥f_{j}∥_{(p)} = ∥f_{j}^{∗}||_{(p)}.  (8.8)

To prove the Lemma, we start with p = 2. In that case (8.4) reduces to (8.1). Let us denote by (8.4)_{p} the inequality (8.4) for a given value of p. We will show

(8.4)_{p} ⇒ (8.4)_{2p}.

This covers only the case p = 2^{k}, but actually the argument used earlier for L_{p}(μ) (see Lemma 3.1) when p is an even integer can be easily adapted to the case of L_{p}(τ) (note that the invariance of I(f_{1}, ⋯, f_{p}) = τ̂(f_{1} ˙⊗ ⋯ ˙⊗f_{p}) under cyclic permutations suffices to adapt this argument here). We leave the details to the reader at this point.

So assume (8.4)_{p} proved for some integer p ≥ 2. Consider

f_{j} ∈ B(H_{j}) ⊗ L_{2p}(τ) j = 1, …, 2p.

Let

g_{j} = f_{2j−1} ˙⊗f_{2j} ∈ B(H_{2j−1} ⊗_{2} H_{2j}) ⊗ L_{p}(τ) (j = 1, …, p).

By (8.4)_{p} we have

∥τ̂(g_{1} ˙⊗ ⋯ ˙⊗g_{p})∥ ≤ ∏_{1}^{p} ∥g_{j}∥_{(p)}.  (8.9)

Moreover using (8.6) we find

∥g_{j}∥_{(p)} = ∥τ̂(f_{2j−1}^{∗} ˙⊗f_{2j−1} ˙⊗f_{2j} ˙⊗f_{2j}^{∗} ˙⊗ ⋯)∥^{\frac{1}{p}}

where the preceding expression is repeated p/2 times. By (8.4)_{p} we have

∥g_{j}∥_{(p)} ≤ ∥f_{2j−1}^{∗} ˙⊗f_{2j−1}∥(p)^{\frac{1}{2}}∥f_{2j} ˙⊗f_{2j}^{∗}∥(p)^{\frac{1}{2}}

and hence by (8.7)

≤ ∥f_{2j−1}∥_{(2p)}∥f_{2j}∥_{(2p)}.

Thus we find that (8.9) implies (8.4)_{2p}.

We then have just like in the commutative case:

Theorem 8.3. Let p ≥ 2 be an even integer. The space L_{p}(τ) can be equipped with an o.s.s. so that denoting by Λ_{p}(τ) the resulting operator space we have for any H and any f in B(H) ⊗ L_{p}(τ)

∥f∥_{B(H)⊗_{min}Λ_{p}(τ)} = ∥f∥_{(p)}.

Proof. We may reduce consideration to H = ℓ_{2} for simplicity of notation. We have then by (8.4)

∥f∥_{(p)} = sup ∥τ̂(f ˙⊗f_{2} ˙⊗ ⋯ ˙⊗f_{p})∥  (8.10)

where the supremum runs over all f_{j} in B(H) ⊗ L_{p}(τ) (2 ≤ j ≤ p) with ∥f_{j}∥_{(p)} ≤ 1. We then define for any x in L_{p}(τ)

J(x) = ⊕[τ̂(x ˙⊗f_{2} ˙⊗ ⋯ ˙⊗f_{p})]  (8.11)

where the direct sum runs over all choices of (f_{j}) (j ≥ 2) as before. Then (8.10) ensures that

∥f∥_{(p)} = ∥(id_{B(H)} ⊗ J)(f)∥_{min}.

Thus J defines an isometric embedding of L_{p}(τ) into some B(H) (here H is a suitably “huge” direct sum) as in (8.11) so that the associated o.s.s. satisfies the desired property.

By exactly the same argument as for Corollary 3.7 above, we have

Corollary 8.4. Let p ≥ q ≥ 2 be even integers. If τ(1) = 1, the inclusion Λ_{p}(τ) ⊂ Λ_{q}(τ) is a complete contraction from Λ_{p}(τ) to Λ_{q}(τ).

It is important for the sequel to observe that 0 ≺ τ̂(f^{∗} ˙⊗f) for any f in B ⊗ L_{2}(τ). This follows from a very general fact on sesquilinear forms.

Lemma 8.5. Let B and E be complex vector spaces. Let x ∈ (B ⊗E)⊗(\overline{B ⊗ E}) be such that x ≻ 0, meaning by this that x can be written as a finite sum x = ∑ t_{k}⊗t̄_{k} with t_{k} ∈ B⊗E. We will use the natural identification \overline{B ⊗ E} = \overline{B}⊗\overline{E}. Let φ: E ⊗ \overline{E} → ℂ be a bilinear form (equivalently φ defines a sesquilinear form on E × E). Let y = (φ ⊗ id_{B⊗\overline{B}})(x) ∈ B ⊗ \overline{B} (more precisely here φ acts on the second and fourth factors, so, to indicate this, the notation y = (φ)_{24}(x) would be less abusive). If φ is positive definite (meaning that φ(a ⊗ ā) ≥ 0 ∀a ∈ E), then y ≻ 0.

Proof. Note that we may as well assume B and E finite dimensional. Consider then t = ∑ b_{k} ⊗ a_{k} ∈ B ⊗ E, ξ ∈ B^{∗} and s = (ξ ⊗ id_{E})(t) ∈ E. We have (ξ ⊗ ξ̄⊗ id_{E⊗\overline{E}})(t ⊗ t̄) = s ⊗ s̄ ≻ 0 and hence (ξ ⊗ ξ̄)(y) = φ(s ⊗ s̄) ≥ 0. By the proof of Lemma 2.1 we conclude that y ≻ 0.

In particular, since τ(a^{∗}a) = τ(aa^{∗}) ≥ 0 for any a in L_{2}(τ), this implies:

Lemma 8.6. For any f in B ⊗ L_{2}(τ), we have

τ̂(f^{∗} ˙⊗f) ≻ 0 (and τ̂(f ˙⊗f^{∗}) ≻ 0).

Remark 8.7. By the classical property of conditional expectations, if 1 ≤ p, p′ ≤ ∞ are conjugate (i.e. p′ = p(p − 1)^{−1}) and if T: L_{p}(τ) → L_{p}(τ) is the conditional expectation with respect to a (von Neumann) subalgebra of M, then: ∀x ∈ L_{p}(τ) ∀y ∈ L_{p′}(τ) we have

τ(T (x)y) = τ(xT (y)) = τ(T (x)T (y)).

Therefore for any f ∈ B(H_{1}) ⊗ L_{p}(τ) and g ∈ B(H_{2}) ⊗ L_{p′}(τ) we have:

τ̂(T (f) ˙⊗g) = τ̂(f ˙⊗T (g)) = τ̂(T (f) ˙⊗T (g))  (8.12)

where we still denote abusively by T the operator I⊗T acting either on B(H_{1})⊗ L_{p}(τ) or on B(H_{2}) ⊗ L_{p′}(τ). Moreover, it is easy to check that T (f^{∗}) = T (f)^{∗} for any f ∈ B ⊗ L_{p}(τ). In the rest of this section we continue to abusively denote by T the operator I ⊗ T on B ⊗ L_{p}(τ).

Lemma 8.8. Let T: L_{p}(τ) → L_{p}(τ) be the conditional expectation with respect to a von Neumann subalgebra N ⊂ M. Let p = 2m be an even integer. Then for any f in B ⊗ L_{p}(τ) we have

∥T f∥_{(p)} ≤ ∥f∥_{(p)}.  (8.13)

Proof. By (8.12), we have

τ̂(T (f) ˙⊗T (f)^{∗} ˙⊗ ⋯ ˙⊗T (f) ˙⊗T (f)^{∗}) = τ̂(f ˙⊗T (f)^{∗} ˙⊗ ⋯ ˙⊗T (f) ˙⊗T (f)^{∗}).

Indeed, just observe that if g = T (f)^{∗} ˙⊗ ⋯ ˙⊗T (f) ˙⊗T (f)^{∗} then T (g) = g. Therefore by (8.4) we have

∥T (f)∥_{(p)}^{p} ≤ ∥f∥_{(p)}∥T (f)∥_{(p)}^{p−1}

and hence after a suitable division we obtain (8.13).

Remark 8.9. In the preceding situation for any f in B ⊗ L_{2}(τ), let f_{0} = T (f) and d_{1} = f − T (f). We have then T (f^{∗} ˙⊗f) = f_{0}^{∗} ˙⊗f_{0} + T (d_{1}^{∗} ˙⊗d_{1}), and hence (since τ̂T = τ̂) τ̂(f^{∗} ˙⊗f) = τ̂(f_{0}^{∗} ˙⊗f_{0}) + τ̂(d_{1}^{∗} ˙⊗d_{1}) Therefore, by Lemma 8.6, we have both τ̂(f_{0}^{∗} ˙⊗f_{0}) ≺ τ̂(f^{∗} ˙⊗f) and τ̂(d_{1}^{∗} ˙⊗d_{1}) ≺ τ̂(f^{∗} ˙⊗f).

By Corollary 8.4, assuming τ(1) = 1, for all even integers p ≥ q ≥ 2, and any f ∈ B ⊗ M, we have ∥f∥_{(p)} ≤ ∥f∥_{(q)}, so that it is again natural to define

∥f∥_{(∞)} = lim_{p→∞} ∥f∥_{(p)}.

This norm is clearly associated to a well defined o.s.s. on M, so we are led to the following

Definition 8.10. Assume τ(1) = 1. We will denote by Λ_{∞}(M, τ) the Banach space M equipped with the o.s.s. determined by the identities

∀f ∈ B ⊗ M ∥f∥_{B⊗_{min}Λ_{∞}(M,τ)} = ∥f∥_{(∞)} = sup_{p∈2ℕ} ∥f∥_{(p)}.

We warn the reader that in sharp contrast with the commutative case, in general Λ_{∞}(M, τ) is not completely isometric to M. See §10 below for more on this, including the case study of M = M_{n} equipped with its normalized trace.

9 Comparisons

We need to recall the definition of the “opposite” of an operator space E ⊂ B(H). The “opposite” of E, denoted by E^{op}, is the same Banach space as E, but equipped with the following norms on M_{n}(E). For any (a_{ij}) in M_{n}(E) we define

∥(a_{ij})∥_{M_{n}(E^{op})} =^{def}∥(a_{ji})∥_{M_{n}(E)}.

Equivalently, E^{op} can be defined as the operator space structure on E for which the transposition: x → ^{t}x ∈ B(H^{∗}) defines a completely isometric embedding of E^{op} into B(H^{∗}). Let M be a von Neumann algebra equipped with a normal semi-finite faithful trace τ, and let L_{p}(τ) be the associated “non-commutative” L_{p}-space. We need to recall the definition of the “natural” o.s.s. on L_{p}(τ) in the sense of [23] (we follow the clarification in [25, p. 139] that is particularly important at this point). We set L_{∞}(τ) = M. Of course we view L_{∞}(τ) as an operator space completely isometric to M. The space L_{1}(τ) is classically defined as the completion of {x ∈ M | τ(|x|) < ∞} for the norm x ↦ ∥x∥_{1} = τ(|x|). It can be identified isometrically with M_{∗} via the mapping x ↦ φ_{x} defined by φ_{x}(a) = τ(xa). The space M_{∗} ⊂ M^{∗} is equipped with the o.s.s. induced by the dual of the von Neumann algebra M (this duality uses Ruan’s theorem, see e.g. [25, 11]). The “natural” o.s.s. on L_{1}(τ) is defined as the one transferred from the space M_{∗}^{op} via the preceding isometric identification x ↦ φ_{x}. In short we declare that L_{1}(τ) = M_{∗}^{op} completely isometrically. Then using complex interpolation, we define the “natural” o.s.s. on L_{p}(τ) (1 < p < ∞) by the completely isometric identity L_{p}(τ) = (L_{∞}(τ), L_{1}(τ))_{1/p}. For example, when (M, τ) = (B(ℓ_{2}), tr), the space L_{p}(τ) can be identified with the Schatten p-class. The column (resp. row) matrices in B(ℓ_{2}) form an operator space usually denoted by C (resp. R). However, when considered as a subspace of L_{1}(τ) they are completely isometric to R (resp. C), while when considered as subspaces of L_{2}(τ) they both are completely isometric to OH. The non-commutative case of §7 requires us to introduce yet another o.s.s. on L_{p}(τ). We set again L_{∞}(τ) = M but we set L_{1}(τ) = M_{∗} (so that L_{1}(τ) = L_{1}(τ)^{op}) and we denote by L_{p}(τ) the operator space defined by

L_{p}(τ) = (L_{∞}(τ), L_{1}(τ))_{1/p}.

The space L_{p}(τ) is isometric to L_{p}(τ) but in the non-commutative case its o.s.s. is different. For instance if (M, τ) = (B(ℓ_{2}), tr), the column (resp. row) matrices in L_{p}(τ) form an operator space that is completely isometric to C (resp. R), for all 1 ≤ p ≤ ∞. In sharp contrast, the o.s.s. of the subspace formed of the diagonal matrices is the same in L_{p}(τ) or L_{p}(τ), and it can be identified completely isometrically with ℓ_{p} equipped with its natural o.s.s. . In particular, L_{2}(τ) is isometric to the Hilbert-Schmidt class S_{2}, but the column (resp. row) matrices in L_{2}(τ) are completely isometric to C (resp. R) while the diagonal ones are completely isometric to OH.

Proposition 9.1. Let 1 ≤ p, q, r ≤ ∞ be such that r^{−1} = p^{−1} + q^{−1}. The product mapping

(x, y) ↦ xy

is (jointly) completely contractive from L_{p}(τ) × L_{q}(τ) to L_{r}(τ).

Proof. We start by the two cases p = r = 1, q = ∞ and q = r = 1, p = ∞. We need to show that (x, φ_{y}) ↦ φ_{xy} (resp. (φ_{x}, y) ↦ φ_{xy}) are (jointly) completely contractive from M × M_{∗} to M_{∗} (resp. from M_{∗} × M to M_{∗}). Consider x = [x_{kl}] ∈ M_{m}(M) with ∥x∥ ≤ 1 and y = [y_{ij}] ∈ M_{n}(M_{∗}) with ∥y∥ ≤ 1 (resp. x = [x_{kl}] ∈ M_{m}(M_{∗}) with ∥x∥ ≤ 1 and y = [y_{ij}] ∈ M_{n}(M) with ∥y∥ ≤ 1). It suffices to show that in both cases we have ∥[φ_{x_{kl}y_{ij}}]∥_{M_{mn}(M_{∗})} ≤ 1. Equivalently, we need to show that the map u : M → M_{n} ⊗ M_{m} defined by u(a) = ∑ e_{ij} ⊗ e_{kl} τ(y_{ij}ax_{kl}) satisfies ∥u∥_{cb} ≤ 1. Consider v : M → M_{m} ⊗ M defined by v(a) = ∑ e_{kl} ⊗ ax_{kl} = (I ⊗ a)x. Clearly ∥v∥_{cb} ≤ ∥x∥ ≤ 1. Let w : M → M_{n} defined by w(a) = ∑ e_{ij}τ(y_{ij}a). Then ∥w∥_{cb} = ∥y∥_{M_{n}(M_{∗})} ≤ 1. We have (I ⊗ w)v(a) = ∑ e_{kl} ⊗ e_{ij} τ(y_{ij}ax_{kl}) which is u(a) up to permutation of the tensor product. Therefore ∥u∥_{cb} ≤ ∥I ⊗ w∥_{cb}∥v∥_{cb} ≤ ∥w∥_{cb}∥v∥_{cb} ≤ 1 (resp. let V : M → M_{n} ⊗ M and W : M → M_{m} be defined by V (a) = ∑ e_{ij} ⊗ ay_{ij} and W (a) = ∑ e_{kl}τ(x_{kl}a), then we have u(a) = (I ⊗ W)V (a) and we conclude similarly).

Corollary 9.2. For any p ∈ 2ℕ and any f ∈ B ⊗ L_{p}(τ) we have

∥f∥_{(p)} = ∥f∥_{B⊗Λ_{p}(τ)} ≤ max{∥f∥_{B⊗_{min}L_{p}(τ)}, ∥f^{∗}∥_{B̄⊗_{min}L_{p}(τ)}}.

In other words, the identity defines a completely contractive map L_{p}(τ) ∩ L_{p}(τ)^{op} → Λ_{p}(τ) (where L_{p}(τ) ∩ L_{p}(τ)^{op} denotes the o.s.s. on L_{p}(τ) induced by the embedding x ↦ x ⊕ x ∈ L_{p}(τ) ⊕ L_{p}(τ)^{op}.

Proof. By iteration, the preceding statement implies that for any integer N the product mapping is (jointly) completely contractive from L_{p_{1}}(τ)×⋯×L_{p_{N}} (τ) to L_{r}(τ) when 1/r = ∑ 1/p_{j}. Equivalently, setting B_{1} = ⋯ = B_{N} = B, the mapping (f_{1}, ⋯, f_{N}) ↦ f_{1} ˙⊗ ⋯ ˙⊗f_{N} is contractive from B_{1} ⊗_{min} L_{p_{1}}(τ) × ⋯ × B_{N} ⊗_{min} L_{p_{N}} (τ) to B_{1} ⊗_{min} ⋯ ⊗_{min} B_{N} ⊗_{min} L_{r}(τ). A fortiori, when r = 1, (f_{1}, ⋯, f_{N}) ↦ τ̂(f_{1} ˙⊗ ⋯ ˙⊗f_{N}) is contractive from B_{1} ⊗_{min} L_{p_{1}}(τ) × ⋯×B_{N} ⊗_{min} L_{p_{N}} (τ) to B_{1} ⊗_{min} ⋯⊗_{min} B_{N}. Therefore, if p is an even integer, we have

∥f∥_{(p)}^{p} = ∥τ̂(f^{∗} ˙⊗f ˙⊗ ⋯ ˙⊗f^{∗} ˙⊗f)∥ ≤ ∥f∥_{B⊗_{min}L_{p}(τ)}^{p/2}∥f^{∗}∥_{B̄⊗_{min}L_{p}(τ)}^{p/2}.

A fortiori we obtain the announced result. Note that x̄ ↦ x^{∗} is a completely isometric linear isomorphism both from M̄ to M^{op} and from M̄_{∗} to M_{∗}^{op}, and hence also from \overline{L_{p}(τ)} to L_{p}(τ)^{op} for all 1 ≤ p ≤ ∞. Therefore, if f = ∑ b_{j} ⊗x_{j} we have ∥f^{∗}∥_{B̄⊗_{min}L_{p}(τ)} = ∥ ∑ b̄_{j} ⊗x_{j}^{∗}∥_{B̄⊗_{min}L_{p}(τ)} = ∥ ∑ b̄_{j} ⊗x̄_{j}∥_{B̄⊗_{min}\overline{L_{p}(τ)^{op}}} = ∥ ∑ b_{j} ⊗ x_{j}∥_{B⊗_{min}L_{p}(τ)^{op}}. Thus ∥f^{∗}∥_{B̄⊗_{min}L_{p}(τ)} = ∥f∥_{B⊗_{min}L_{p}(τ)^{op}}, whence the last assertion.

We will now examine the particular case when (M, τ) = (B(ℓ_{2}), tr). Recall that R (resp. C) is the subspace of M = B(ℓ_{2}) formed by all row (resp. column) matrices. More generally, we denote by R_{p} (resp. C_{p}) the operator space obtained by equipping R (resp. C) with the o.s.s. induced by L_{p}(τ). We also denote by R_{p}^{n} (resp. C_{p}^{n}) the n-dimensional version of R_{p} (resp. C_{p}). Similarly, we will denote by R̃_{p} (resp. C̃_{p}) the operator space obtained by equipping R (resp. C) with the o.s.s. induced by Λ_{p}(τ). Furthermore, let D̃_{p} be the operator subspace of Λ_{p}(τ) formed of all the diagonal matrices. As a Banach space this is isometric to ℓ_{p}, and it is easy to check that as an operator space D̃_{p} is completely isometric to the space λ_{p} = Λ_{p}(ℕ, μ) with μ equal to the counting measure on ℕ. Let b_{j} ∈ B (j = 1, ⋯, n) and let f = ∑ b_{j} ⊗e_{1j} ∈ B⊗R (resp. g = ∑ b_{i}⊗e_{i1} ∈ B ⊗ C). Then f ˙⊗f^{∗} = ∑ b_{j} ⊗ b̄_{j} ⊗ e_{11} (resp. g^{∗} ˙⊗g = ∑ b̄_{j} ⊗ b_{j} ⊗ e_{11}). Note that ∥ ∑ b̄_{j} ⊗ b_{j}∥^{1/2} = ∥ ∑ b_{j} ⊗ b̄_{j}∥^{1/2}. Therefore, viewing f and g as elements of B ⊗ Λ_{p}(τ), for any p ∈ 2ℕ, we have

∥f∥_{(p)} = ∥g∥_{(p)} = ∥ ∑ b_{j} ⊗ b̄_{j}∥^{1/2}.

Thus we find:

Lemma 9.3. The spaces R̃_{p} and C̃_{p} are both completely isometric to OH for any p ∈ 2ℕ, while D̃_{p} is completely isometric to λ_{p}.

Again let b_{j} ∈ B (j = 1, ⋯, n) and let f = ∑ b_{j} ⊗ e_{1j}. We have ∥f∥_{B⊗L_{p}(τ)} = sup{∥ ∑ b_{j}ab_{j}^{∗}∥_{p}^{1/2} | ∥a∥_{p} ≤ 1} (see [22, p. 83-84] or [36] for details). In case b_{j} = e_{j1}, this gives us ∥f∥_{B⊗L_{p}(τ)} = n^{1/2p}. Therefore the natural inclusion R_{p}^{n} → R_{2}^{n} has c.b. norm ≥ n^{1/4−1/2p}. Similarly, using instead b_{j} = e_{1j}, we find ∥f∥_{B⊗L_{p}(τ)} = n^{(1/2)(1−1/p)} and hence ∥R_{2}^{n} → R_{p}^{n}∥_{cb} ≥ n^{1/4−1/2p}. This shows:

Lemma 9.4. For any p ∈ 2ℕ and any integer n ≥ 1, the n-dimensional identity maps satisfy

∥L_{p}(M_{n}, tr) → Λ_{p}(M_{n}, tr)∥_{cb} ≥ n^{1/4−1/2p}

and

∥Λ_{p}(M_{n}, tr) → L_{p}(M_{n}, tr)∥_{cb} ≥ n^{1/4−1/2p}.

10 Connection with CB maps on OH

Given a Hilbert space H we denote by OH the operator Hilbert space isometric to H, as defined in [22]. This means that whenever (T_{j}) is an orthonormal basis of OH, for any finitely supported family (b_{j}) in B we have

∥ ∑ b_{j} ⊗ T_{j}∥ = ∥ ∑ b̄_{j} ⊗ b_{j}∥^{1/2}.  (10.1)

Assume M ⊂ B(H) and τ(1) = 1. We will compare the limit o.s.s. of Λ_{p}(M, τ) when p → ∞ to the one induced on M by CB(OH) equipped with its usual operator space structure. The latter can be described as follows (see e.g. [11]): Whenever E, F are operator spaces the space CB(E, F) of all c.b. maps from E to F is equipped with the (unique) o.s.s. determined by the isometric identity

∀N ≥ 1 M_{N}(CB(E, F)) = CB(E, M_{N}(F)).

More generally, we have an isometric embedding

B ⊗_{min} CB(E, F) ⊂ CB(E, B ⊗_{min} F).  (10.2)

If either E or F is finite dimensional, we may identify completely isometrically CB(E, F) with E^{∗} ⊗_{min} F. When E = F, we denote simply CB(E) = CB(E, E). Thus in particular CB(OH_{n}) can be identified with OH_{n}^{∗}⊗_{min}OH_{n}, or equivalently by the selfduality of OH_{n}, with \overline{OH_{n}}⊗_{min}OH_{n} or OH_{n} ⊗_{min} \overline{OH_{n}}. We first recall a well known fact.

Lemma 10.1. Let E, F, G be operator spaces. Let B′ = B(H′) for some Hilbert space H′. Then for any f = ∑ b_{j} ⊗ x_{j} ∈ B ⊗ CB(F, G) and g = ∑ b′_{k} ⊗ y_{k} ∈ B′ ⊗ CB(E, F) we have

∥f ˙⊗g∥_{B⊗_{min}B′⊗_{min}CB(E,G)} ≤ ∥f∥_{B⊗_{min}CB(F,G)} ∥g∥_{B′⊗_{min}CB(E,F)}, (10.3)

where, as before, we denote f ˙⊗g = ∑_{j,k} b_{j} ⊗ b′_{k} ⊗ x_{j}y_{k} ∈ B ⊗ B′ ⊗ CB(E, G). In other words, the composition (x, y) ↦ xy is (jointly) completely contractive from CB(F, G) × CB(E, F) to CB(E, G).

Proof. To prove (10.3), note that f (resp. g) defines a c.b. map f̃ : F → G ⊗_{min} B (resp. g̃ : E → F ⊗_{min} B′) and ∥f∥_{B⊗_{min}CB(F,G)} = ∥f̃∥_{cb} (resp. ∥g∥_{B′⊗_{min}CB(E,F)} = ∥g̃∥_{cb}). Indeed, recall that, if we wish, G ⊗_{min} B can be identified with B ⊗_{min} G. Similarly, f ˙⊗g defines a c.b. map Ψ : E → G ⊗_{min} B ⊗_{min} B′ such that ∥f ˙⊗g∥_{B⊗_{min}B′⊗_{min}CB(E,G)} = ∥Ψ∥_{cb}. But since Ψ = (f̃⊗ Id_{B′}) ∘ g̃ we have ∥Ψ∥_{cb} ≤ ∥f̃∥_{cb}∥g̃∥_{cb} and (10.3) follows.

Remark 10.2. In particular, the preceding Lemma implies a fortiori that if D, E, F, G are operator spaces and if u ∈ CB(D, E) and v ∈ CB(F, G) are fixed complete contractions, then the mapping x ↦ vxu is a complete contraction from CB(E, F) to CB(D, G). Indeed, the latter can be viewed as the restriction of the triple product map to ℂv × CB(E, F) × ℂu.

Theorem 10.3. Let (M, τ) be as before with M ⊂ B(H) and τ(1) = 1. Let us denote by \underline{M} the operator space obtained by equipping M with the o.s.s. induced by CB(OH). Then

Λ_{∞}(M, τ) = \underline{M}

completely isometrically.

The proof of this Theorem will require some observations about the space CB(OH) that may be of independent interest. The following rather striking identity (10.4) appears as analogous to Gelfand’s axiom (namely ∥x∥^{2} = ∥x^{∗}x∥) for C^{∗}-algebras. It seems to express that CB(OH) is an o.s. analogue of a C^{∗}-algebra...

Theorem 10.4. Let us denote simply by B the operator space CB(OH). (Note that B is isometric to B(H) as a Banach space.) For any f ∈ B ⊗ B we have

∥f∥_{B⊗_{min}B}^{2} = ∥f^{∗} ˙⊗f∥_{B̄⊗_{min}B⊗_{min}B} = ∥f ˙⊗f^{∗}∥_{B⊗_{min}B̄⊗_{min}B}.  (10.4)

Moreover, we also have

∥f^{∗}∥_{B̄⊗_{min}B} = ∥f∥_{B⊗_{min}B}.  (10.5)

Proof. Let H_{i} ⊂ H be an increasing net of finite dimensional subspaces with dense union. Assuming f = ∑ b_{j} ⊗ x_{j}, let f(i) = ∑ b_{j} ⊗ P_{H_{i}}x_{j}_{|H_{i}} ∈ B ⊗ CB(OH_{i}). Then, using the homogeneity of OH in the sense of [22, p. 19] or [23], one checks that each side of either (10.4) or (10.5) is equal to the supremum over i of the expression obtained after substituting f_{i} for f. Thus it suffices to prove (10.4) or (10.5) when dim(H) < ∞. In that case, denoting by T_{j} an orthonormal basis of OH_{n}, and using the identity CB(OH_{n}) = OH_{n} ⊗_{min} \overline{OH_{n}}, we may write any f ∈ B ⊗ B as f = ∑ b_{ij} ⊗T_{i}⊗T̄_{j} with b_{ij} ∈ B, and ∥f∥_{B⊗B} = ∥ ∑ b_{ij} ⊗T_{i}⊗T̄_{j}∥_{B⊗_{min}OH_{n}⊗_{min}\overline{OH_{n}}}. Using ∥x∥ = ∥x̄∥ for any operator x, and permuting the second and third factors, we have then obviously (the norm being the min-norm)

∥ ∑ b_{ij}⊗T_{i}⊗T̄_{j}∥ = ∥ ∑ b̄_{ij}⊗T̄_{i}⊗T_{j}∥ = ∥ ∑ b̄_{ij}⊗T_{j}⊗T̄_{i}∥ = ∥ ∑ b̄_{ji}⊗T_{i}⊗T̄_{j}∥,

and this is clearly equivalent to (10.5). Let y_{i} = ∑_{j} b_{ij} ⊗T̄_{j}. By (10.1), we have ∥ ∑ b_{ij} ⊗T_{i} ⊗T̄_{j}∥_{B⊗_{min}OH_{n}⊗_{min}\overline{OH_{n}}} = ∥ ∑ y_{i} ⊗T_{i}∥ = ∥ ∑ ȳ_{i} ⊗y_{i}∥^{1/2} = ∥ ∑ y_{jk} ⊗T_{j} ⊗T̄_{k}∥^{1/2} where y_{jk} = ∑_{i} b̄_{ij} ⊗b_{ik}. Using again the identity CB(OH_{n}) = OH_{n} ⊗_{min} \overline{OH_{n}} we find f^{∗} ˙⊗f = ∑ y_{jk} ⊗ T_{j} ⊗ T̄_{k}. Thus, we have ∥f∥ = ∥f^{∗} ˙⊗f∥^{1/2}, and by (10.5) we obtain (10.4).

Remark 10.5. Note that after iteration, for any p = 2^{m} (m ≥ 1), (10.4) yields

∥f∥_{B⊗_{min}B}^{p} = ∥f^{∗} ˙⊗f ˙⊗f^{∗} ˙⊗f ⋯ ∥_{B̄⊗_{min}B⊗_{min}···⊗_{min}B}.  (10.6)

Corollary 10.6. Let H_{I} = ⊕_{i∈I}H_{i} be an orthogonal decomposition of a Hilbert space H_{I}. We have then a completely isometric embedding

⊕_{i∈I}CB(OH_{i}) ⊂ CB(OH_{I}).

Proof. Let u : ⊕_{i∈I}CB(OH_{i}) → CB(OH_{I}) denote this embedding. It is easy to reduce the proof to the finite case so we assume |I| < ∞. Since the coordinatewise inclusions and projections relative to OH_{I} are all completely contractive, it is easy to check using Lemma 10.1 that ∥u∥_{cb} ≤ |I| < ∞. Consider now f ∈ B ⊗_{min} (⊕_{i∈I}CB(OH_{i})) and let g = (Id_{B} ⊗ u)(f) ∈ B ⊗_{min} CB(OH_{I}). We need to show that ∥g∥ = ∥f∥. Note that (Id_{B̄⊗B⊗···} ⊗ u)(f^{∗} ˙⊗f)^{˙⊗m}) = (g^{∗} ˙⊗g)^{˙⊗m}. Thus, by (10.3) and (10.4) we have for any integer m

∥g∥ = ∥(g^{∗} ˙⊗g)^{˙⊗m}∥^{1/2m} ≤ (|I|∥(f^{∗} ˙⊗f)^{˙⊗m}∥)^{1/2m} = |I|^{1/2m}∥f∥

so that letting m → ∞ we obtain ∥g∥ ≤ ∥f∥. Since the converse inequality follows easily from Remark 10.2 applied to the coordinate projections, we have equality.

Theorem 10.7. Let E ⊂ B(H) be any operator space. Let us denote again by \underline{E} the operator space obtained by inducing on E the o.s.s. of CB(OH). Let F ⊂ B(K) be another operator space. Then for any u ∈ CB(E, F) we have

∥u∥_{CB(\underline{E},\underline{F})} ≤ ∥u∥_{CB(E,F)}.

In particular, if u : E → F is completely isometric, then u : \underline{E} → \underline{F} also is.

Proof. We may clearly assume F = B(K) and \underline{F} = CB(OK) and by the same argument as in the preceding proof, we may assume dim(K) = n < ∞. Assume ∥u∥_{CB(E,F)} ≤ 1. Then u extends to a c.b. map û : B(H) → B(K) with the same cb-norm. Since M_{n}(B(H)^{∗}) = M_{n}(B(H)_{∗})^{∗∗} isometrically (see e.g. [11, p. 75]) û is then a point norm limit of normal maps with cb-norm ≤ 1, so we may assume that u is normal on E = B(H). Then (see [34, p. 45]) there is a factorization of u of the form u(x) = V ρ(x)W with ∥V ∥ ≤ 1, ∥W ∥ ≤ 1 where ρ is an “ampliation”, i.e. ρ takes its values in B(⊕_{i∈I}H_{i}) for some set I with H_{i} = H for all i ∈ I and ρ(x) = ⊕_{i∈I}ρ_{i}(x) with ρ_{i}(x) = x for all i ∈ I. This reduces the Lemma to the case when u is an ampliation and to the case when u is of the form u(x) = V xW. Let us first assume u(x) = V xW with V : H → K and W : K → H of norm 1. By the homogeneity of OH we know that the cb norm of V : OH → OK is 1, and similarly for W : OK → OH. Then by Remark 10.2 ∥u∥_{CB(CB(OH),CB(OK))} ≤ 1. We now assume that u is an ampliation i.e. u = u_{I} where u_{I}(x) = ⊕_{i∈I}u_{i}(x) ∈ B(⊕_{i∈I}H_{i}) with H_{i} = H and u_{i}(x) = x for all i ∈ I. Let H_{I} = ⊕_{i∈I}H_{i}. By Corollary 10.6 u_{I} is a complete isometry from CB(OH) to CB(OH_{I}). Since both multiplications and ampliations have been checked, the proof of the first assertion is complete. The second assertion is then immediate.

Corollary 10.8. Let E ⊂ B(H) be any operator space. The o.s.s. of \underline{E} (induced on E by that of CB(OH)) is independent of the completely isometric embebdding E ⊂ B(H), i.e. it depends only on the o.s.s. of E.

Proof of Theorem 10.3. We first give a simple argument for the special case when M = M_{n} equipped with its normalized trace τ_{n}. We will show that Λ_{∞}(M_{n}, τ_{n}) can be identified completely isometrically with CB(OH_{n}) for any n ≥ 1. We first claim that the identity from \overline{OH_{n}} ⊗ OH_{n} to itself induces a mapping V_{n} : \overline{OH_{n}} ⊗_{h} OH_{n} → \overline{OH_{n}} ⊗_{min} OH_{n} such that ∥V_{n}∥_{cb} ≤ 1 and ∥V_{n}^{−1}∥_{cb} ≤ n^{1/2}. By the minimality of the minimal tensor product the first assertion is obvious. To check the second one, recall the identity map on n-dimensional Hilbert space defines an isomorphism u_{n} : R_{n} → OH_{n} such that ∥u_{n}∥_{cb} = ∥u_{n}^{−1}∥_{cb} = n^{1/4} (see e.g. [25, p. 219]). Therefore, we have a factorization of V_{n}^{−1} as follows

\overline{OH_{n}} ⊗_{min} OH_{n} →^{Id⊗u_{n}^{−1}}\overline{OH_{n}} ⊗_{min} R_{n} = \overline{OH_{n}} ⊗_{h} R_{n} →^{Id⊗u_{n}}\overline{OH_{n}} ⊗_{h} OH_{n},

where we used the identity E ⊗_{min} R_{n} = E ⊗_{h} R_{n} for which we refer e.g. to [25, p. 95]. From this follows ∥V_{n}^{−1}∥_{cb} ≤ ∥u_{n}^{−1}∥_{cb}∥u_{n}∥_{cb} = n^{1/2}. More explicitly, recall that for any pair of Hilbert spaces H, K we have (see [22, Cor. 2.12])

OH ⊗_{h} OK = O(H ⊗_{2} K);

in particular, Λ_{2}(M_{n}, tr) is the same as L_{2}(M_{n}, tr) = \overline{OH_{n}} ⊗_{h} OH_{n}. Therefore, for any g ∈ B ⊗ Λ_{2}(M_{n}, tr) we have

∥g∥_{B⊗_{min}CB(OH_{n})} ≤ ∥g∥_{B⊗_{min}Λ_{2}(M_{n},tr)} ≤ n^{1/2}∥g∥_{B⊗_{min}CB(OH_{n})}. (10.7)

Consider an even integer p and f ∈ B ⊗ M_{n}. Let g = f^{∗} ˙⊗f ˙⊗ ⋯ (here there are p/2 factors equal to f^{∗} ˙⊗f). We have by (10.4)

∥g∥_{B̄⊗_{min}···⊗_{min}CB(OH_{n})}^{2} = ∥g ˙⊗g^{∗}∥_{B̄⊗_{min}···⊗_{min}CB(OH_{n})}.

We have also g ˙⊗g^{∗} = f^{∗} ˙⊗f ˙⊗ ⋯ (here there are p such factors) and hence, assuming p = 2^{m} for some m and using (10.6), we find

∥g∥_{B̄⊗_{min}···⊗_{min}CB(OH_{n})}^{2} = ∥f∥_{B⊗_{min}CB(OH_{n})}^{p}.  (10.8)

By definition of Λ_{p} we have

∥f∥_{B⊗_{min}Λ_{p}(M_{n},tr)}^{p} = ∥g∥_{Λ_{2}(M_{n},tr)}^{2}

and hence by (10.7)

∥g∥_{B⊗_{min}CB(OH_{n})}^{2} ≤ ∥f∥_{B⊗_{min}Λ_{p}(M_{n},tr)}^{p} ≤ n∥g∥_{B⊗_{min}CB(OH_{n})}^{2}.

Then by (10.8) we obtain

∥f∥_{B⊗_{min}CB(OH_{n})}^{p} ≤ ∥f∥_{B⊗_{min}Λ_{p}(M_{n},tr)}^{p} ≤ n∥f∥_{B⊗_{min}CB(OH_{n})}^{p},

and, if we take the p-th root and let p → ∞ this yields

∥f∥_{B⊗_{min}CB(OH_{n})} = ∥f∥_{B⊗_{min}Λ_{∞}(M_{n},τ_{n})}.

We now consider the general case. Let f ∈ B ⊗ M. For any p ∈ 2ℕ we have ∥f∥_{(p)}^{p} = τ̂(f^{∗} ˙⊗f ⋯). As a linear form on M, τ has norm 1, and hence c.b. norm equal to 1 on \underline{M}. Therefore

∥f∥_{(p)}^{p} = ∥τ̂(f^{∗} ˙⊗f ⋯)∥ ≤ ∥f^{∗} ˙⊗f ⋯ ∥_{B̄⊗···B⊗\underline{M}} = ∥f^{∗} ˙⊗f ⋯ ∥_{B̄⊗···B⊗B}

and by (10.6) (assuming p = 2^{m})

∥f^{∗} ˙⊗f ⋯ ∥_{B̄⊗···B⊗B} = ∥f∥_{B⊗_{min}B}^{p} = ∥f∥_{B⊗_{min}\underline{M}}^{p}.

Thus we obtain ∥f∥_{(p)} ≤ ∥f∥_{B⊗_{min}\underline{M}}, and taking the supremum over p yields

∥f∥_{B⊗_{min}Λ_{∞}(M,τ)} ≤ ∥f∥_{B⊗_{min}\underline{M}}.

It remains to prove the converse inequality. Consider f ∈ B ⊗ M. Let F : OH → OH ⊗_{min} B be the associated c.b. map (as in the proof of Lemma 10.1). By Corollary 10.8 we may assume that H = L_{2}(τ) and that the inclusion M ⊂ B(L_{2}(τ)) is the usual realization of M acting on L_{2}(τ) by left multiplication. Let B′ be another copy of B. Note that for any ξ ∈ B′ ⊗ OH we have

∥ξ∥_{B′⊗_{min}OH} = ∥ξ∥_{(2)}.

Moreover, if ξ ∈ B′ ⊗ M ⊂ B′ ⊗ OH, then up to permutation of factors (Id_{B′} ⊗ F)(ξ) ≈ f ˙⊗ξ. Since ∥F ∥_{cb} = ∥Id_{B′} ⊗ F ∥_{cb}, the definition of the o.s.s of CB(OH) (see (10.2) above) shows that

∥f∥_{B⊗_{min}\underline{M}} = ∥F ∥_{cb} = sup\{∥f ˙⊗ξ∥_{(2)} | ξ ∈ B′ ⊗ M, ∥ξ∥_{(2)} ≤ 1\}.

Fix ξ ∈ B′ ⊗ M with ∥ξ∥_{(2)} ≤ 1. To complete the proof it suffices to show that

∥f ˙⊗ξ∥_{(2)} ≤ sup_{p∈2ℕ} ∥f∥_{(p)} = ∥f∥_{B⊗_{min}Λ_{∞}(M,τ)}.

To verify this, we claim that for any p of the form p = 2^{m} we have

∥f ˙⊗ξ∥_{(2)} ≤ ∥τ̂((f^{∗} ˙⊗f)^{˙⊗p/2} ˙⊗ξ ˙⊗ξ^{∗})∥^{1/p}.  (10.9)

This is easy to check by induction on m. Indeed, by (8.3) (for p = 2), equality holds in the case m = 1 and if we assume our claim proved for a given value of m then the Haagerup-Cauchy-Schwarz inequality (8.2) shows that it holds also for m + 1, because we may write (recall (8.1))

∥τ̂((f^{∗} ˙⊗f)^{˙⊗p/2} ˙⊗ξ ˙⊗ξ^{∗})∥ ≤ ∥(f^{∗} ˙⊗f)^{˙⊗p/2} ˙⊗ξ∥_{(2)}∥ξ^{∗}∥_{(2)} =

= ∥τ̂((f^{∗} ˙⊗f)^{˙⊗p} ˙⊗ξ ˙⊗ξ^{∗})∥^{1/2}∥ξ^{∗}∥_{(2)},

and by (8.5) ∥ξ^{∗}∥_{(2)} ≤ 1, so we obtain (10.9) with 2p in place of p. We now use the claim to conclude: By (8.2) again (or by (8.4)) we have

∥τ̂((f^{∗} ˙⊗f)^{˙⊗p/2} ˙⊗ξ ˙⊗ξ^{∗})∥^{1/p} ≤ ∥f∥_{(2p)}∥ξ ˙⊗ξ^{∗}∥_{(2)}^{1/p}.

Now ξ ∈ B′ ⊗ M implies ξ ˙⊗ξ^{∗} ∈ B′ ⊗ B̄′ ⊗ M and, since τ is finite, we have ∥ξ ˙⊗ξ^{∗}∥_{(2)} < ∞, therefore ∥ξ ˙⊗ξ^{∗}∥_{(2)}^{1/p} → 1 when p → ∞ and we deduce from (10.9)

∥f ˙⊗ξ∥_{(2)} ≤ lim sup_{p→∞} ∥τ̂((f^{∗} ˙⊗f)^{˙⊗p/2} ˙⊗ξ ˙⊗ξ^{∗})∥^{1/p} ≤ lim sup_{p→∞} ∥f∥_{(2p)} = sup_{p∈2ℕ} ∥f∥_{(p)},

which completes the proof.

Remark 10.9. By the minimality of the min tensor product, we know that we have a completely contractive inclusion OH^{∗} ⊗_{h} OH → OH^{∗} ⊗_{min} OH ⊂ CB(OH). Therefore, for any pair of sets I, J, in analogy with the inclusion of the Hilbert-Schmidt class into the bounded operators, we have a completely contractive inclusion

OH(I × J) → CB(OH(I), OH(J)).

11 Non-commutative Khintchine inequalities

We start by a fairly simple statement mimicking a classical commutative fact:

Proposition 11.1. Let p = 2n. Let {x_{k}} be a sequence in L_{p}(τ), such that, for some constant C, for any finite sum f = ∑ b_{k} ⊗ x_{k} with coefficients b_{k} in B(H), we have

∥f∥_{(p)} ≤ C ∥∑ b_{k} ⊗ b̄_{k}∥^{1/2} .  (11.1)

Assume moreover that {x_{k}} is orthonormal in L_{2}(τ) and τ(1) = 1. Then the closed span of (x_{k}) in Λ_{p}(τ) is completely isomorphic to OH and completely complemented in Λ_{p}(τ). More precisely the orthogonal projection P onto this span satisfies ∥P: Λ_{p} → Λ_{p}∥_{cb} ≤ C.

Proof. Let P be the orthogonal projection on Λ_{2} onto the span under consideration. For any f ∈ B ⊗ Λ_{p}, let h = (Id ⊗ P)(f). By a well known fact (see [22, p. 19]), P is completely contractive on Λ_{2}, so that ∥h∥_{(2)} ≤ ∥f∥_{(2)}. By Corollary 8.4, we have ∥f∥_{(2)} ≤ ∥f∥_{(p)} and by our assumption ∥h∥_{(p)} ≤ C∥h∥_{(2)}. Therefore ∥h∥_{(p)} ≤ C∥f∥_{(p)}. Thus, the c.b. norm of P acting from Λ_{p} to itself is automatically ≤ C. Moreover, for any h ∈ B ⊗ \overline{span}[x_{k}], we have ∥h∥_{(2)} ≤ ∥h∥_{(p)} ≤ C∥h∥_{(2)}, which shows that the span is completely isomorphic to OH.

With the “natural” o.s.s. introduced in [23] the Khintchine inequalities for 1 < p < ∞ are due to F. Lust-Piquard [20] . For p an even integer, A. Buchholz [7] found a beautiful proof that yields optimal constants. His proof is valid for a much more general class of variables instead of the Rademacher functions. We will now follow his ideas to investigate the analogous question in the space Λ_{p}. Let P_{2}(2n) denote the set of all partitions of [1, …, 2n] onto subsets each with exactly 2 elements. So an element ν in P_{2} can be described as a collection of disjoint pairs {k_{i}, j_{i}} (1 ≤ i ≤ n) with k_{i} ≠ j_{i} such that {1, …, 2n} = {k_{1}, …, k_{n}, j_{1}, …, j_{n}}. We call such a partition into pairs a 2-partition. Let p = 2n be an even integer ≥ 2. Following [6] we say that a sequence {x_{k}} in L_{p}(τ), has p-th moments defined by pairings if there is a function ψ: P_{2}(2n) → ℂ defined on the set of 2-partitions of [2n] = {1, …, 2n} such that for any k_{1}, …, k_{2n} we have

τ(x_{k_{1}}x_{k_{2}}^{∗}x_{k_{3}} … x_{k_{2n−1}}x_{k_{2n}}^{∗}) = ∑_{ν∼(k_{1},...,k_{2n})} ψ(ν)

where the notation ν ∼ (k_{1}, …, k_{2n}) means that k_{i} = k_{j} whenever the pair {i, j} is a block of the partition ν. Note that, for each k, taking the k_{j}’s all equal to k, this implies

τ(|x_{k}|^{p}) = ∑_{ν∈P_{2}(2n)} ψ(ν).  (11.2)

Now let E = span[x_{j}] and B = B(H). Consider f ∈ B ⊗ E of the form

f = ∑ b_{j} ⊗ x_{j}.

We have

τ̂((f ˙⊗ f^{∗})^{˙⊗ n}) = ∑_{k_{1},...,k_{2n}} ∑_{ν∼(k_{1},...,k_{2n})} ψ(ν)b_{k_{1}} ⊗ b̄_{k_{2}} ⊗ ⋯ ⊗ b_{k_{2n−1}} ⊗ b̄_{k_{2n}}.

Therefore

∥f∥_{(2n)}^{2n} = ∥ ∑_{ν∈P_{2}(2n)} ψ(ν) ∑_{(k_{1},...,k_{2n})∼ν} b_{k_{1}} ⊗ b̄_{k_{2}} ⊗ ⋯ ⊗ b̄_{k_{2n}} ∥

≤ ∑_{ν∈P_{2}(2n)} |ψ(ν)| ∥ ∑_{(k_{1},...,k_{2n})∼ν} b_{k_{1}} ⊗ b̄_{k_{2}} ⊗ ⋯ ⊗ b̄_{k_{2n}} ∥ .

But now let

Φ(ν) = ∑_{(k_{1},...,k_{2n})∼ν} b_{k_{1}} ⊗ b̄_{k_{2}} ⊗ ⋯ ⊗ b_{k_{2n−1}} ⊗ b̄_{k_{2n}}.

Then up to permutation Φ(ν) is equal to a product of n terms of the form either ∑ b_{k} ⊗ b_{k}, ∑ b̄_{k} ⊗ b̄_{k} or ∑ b_{k} ⊗ b̄_{k}. Let T_{1}, ⋯ T_{n} be an enumeration of the latter terms. Since the permutation leaves the norm invariant, we have ∥Φ(ν)∥ = ∏_{1}^{n} ∥T_{j}∥. By (3.1) ∥T_{j}∥ ≤ ∥ ∑ b_{k} ⊗ b̄_{k}∥ for each j (actually there is equality for terms the third kind), and hence

∥Φ(ν)∥ ≤ ∥∑ b_{k} ⊗ b̄_{k}∥^{n}

and we conclude that

∥f∥_{(2n)} ≤ ( ∑_{ν∈P_{2}(2n)} |ψ(ν)|)^{1/2n} ∥∑ b_{k} ⊗ b̄_{k}∥^{1/2} .  (11.3)

Moreover by (11.2) we know that if ψ(ν) ≥ 0 for all ν, then the constant ∑_{ν∈P_{2}(2n)} |ψ(ν)| is optimal. Recapitulating, we have proved:

Theorem 11.2. Let p = 2n. Let {x_{k}} be as above a sequence in L_{p}(τ), with p-th moments defined by pairings via a function ψ: P_{2}(2n) → ℂ. Then for any finite sum f = ∑ b_{k} ⊗ x_{k} (b_{k} ∈ B(H)), we have

∥f∥_{(p)} ≤ C_{ψ,p} ∥∑ b_{k} ⊗ b̄_{k}∥^{1/2} ,  (11.4)

where C_{ψ,p} = (∑_{ν∈P_{2}(2n)} |ψ(ν)|)^{1/2n}. Moreover this constant is optimal if ψ(ν) ≥ 0 for all ν.

Buchholz applied the preceding statement to a q-Gaussian family with q ∈ [−1, 1]. The latter have moments defined by pairings. When q ∈ [0, 1], the function ψ is non-negative, so the constant C_{ψ,p} is optimal and, by (11.2), we know C_{ψ,p} = ∥x_{1}∥_{p}. In particular, we have:

Corollary 11.3. Let (x_{k}) be a sequence of independent Gaussian normal random variables on a probability space (Ω, ℙ). Then the span of (x_{k}) is completely isomorphic to OH and is completely complemented in Λ_{p}(Ω, ℙ) for every even integer p. Moreover, (11.4) holds with a constant C_{ψ,p} = ∥x_{1}∥_{p} that is O(\sqrt{p}) when p → ∞.

Remark 11.4. The preceding Corollary also holds when (x_{k}) is a sequence (ε_{k}) of independent symmetric ±1 valued variables (or equivalently for the Rademacher functions). We show this in Corollary 11.12 below, but here is a quick proof with a slightly worse constant. Let (x_{k}) be independent Gaussian normal random variables and assume that (ε_{k}) is independent from (x_{k}). It is well known that (x_{k}) has the same distribution as (ε_{k}|x_{k}|). Let δ = 𝔼(|x_{k}|) = 2/\sqrt{π}. The conditional expectation E with respect to (ε_{k}) satisfies E(ε_{k}|x_{k}|) = δε_{k}. Therefore δ ∑ ε_{k}b_{k} = E(∑ ε_{k}|x_{k}|b_{k}), and by (8.13), this implies

δ∥ ∑ ε_{k}b_{k}∥_{(p)} ≤ ∥ ∑ ε_{k}|x_{k}|b_{k}∥_{(p)} = ∥ ∑ x_{k}b_{k}∥_{(p)}.

So we obtain the Rademacher case with a constant ≤ δ^{−1}C_{ψ,p} since Proposition 11.1 ensures the complete complementation. The preceding result applies to q-Gaussian and in particular free semi-circular (or circular) elements, see [7] for details. We have then C_{ψ,p} ≤ 2/\sqrt{1 − |q|} for all even p. In either the semi-circular (q = 0) or the circular case, we have C_{ψ,p} ≤ 2 for all even p, and hence:

Corollary 11.5. For any even integer p, the closed span of a free semicircular (or circular) family, is completely isomorphic to OH and completely complemented (by the orthogonal projection) in the space Λ_{p} for the associated trace (on the free group factor). Moreover, the corresponding constants are bounded by 2 uniformly over p.

Corollary 11.6. Let M be the von Neumann algebra of the free group 𝔽_{∞} with infinitely many generators (g_{k}). For any p = 2n and any finite sum f̃ = ∑ b_{k} ⊗ λ(g_{k}) (b_{k} ∈ B(H)), we have

∥∑ b_{k} ⊗ b̄_{k}∥^{1/2} ≤ ∥f̃∥_{(p)} ≤ 2 ∥∑ b_{k} ⊗ b̄_{k}∥^{1/2} .  (11.5)

More generally, let W_{d} ⊂ 𝔽_{∞} denote the subset formed of the reduced words of length d. Then for any finitely supported function b : W_{d} → B we have

∥ ∑_{t∈W_{d}} b(t) ⊗ λ(t)∥_{B⊗\underline{M}} ≤ (d + 1) ∥∑_{t∈W_{d}} b(t) ⊗ b̄(t)∥^{1/2} .  (11.6)

Proof. The left hand side of (11.5) follows from Corollary 8.4 with q = 2 and the orthonormality of (λ(g_{k})) in L_{2}(τ). By [5, Th. 2.8] the operator space spanned by {λ(t)}_{t∈W_{d}} is completely isomorphic to the intersection X of a family of d + 1 operator spaces X_{i}, 0 ≤ i ≤ d, with associated constant equal to d + 1. On one hand, the space X_{0} (resp. X_{d}) is completely isometric to R (resp. C), the underlying respective Hilbert space being ℓ_{2}(W_{d}). On the other hand, when 0 < j < d the space X_{j} is completely isometric to the subspace of B(ℓ_{2}(W_{d−j}), ℓ_{2}(W_{j})) associated to matrices of the form [a(st)] when a is supported on W_{d}. Identifying each W_{i} simply with ℕ we see that \underline{X_{0}} (resp. \underline{X_{d}}) is completely isometric to OH(ℕ), while \underline{X_{i}} is completely isometric to the associated subspace of CB(OH(ℕ)). By Remark 10.9 we have a completely isometric inclusion \underline{X_{0}} → \underline{X_{i}} for any 0 ≤ i ≤ d, therefore the intersection of the family \underline{X_{i}} 0 ≤ i ≤ d is completely isometric to OH with H = ℓ_{2}(W_{d}). Since by Corollary 10.6 we know that \underline{X} = ∩_{0≤i≤d}\underline{X_{i}}, (11.6) follows.

Remark 11.7. A comparison with known results (see [7] for detailed references) shows that the limit of ∥f̃∥_{(p)} when p → ∞ is not equivalent to ∥f̃∥_{B(H)⊗_{min}M} (here M is the von Neumann algebra of the free group with infinitely many generators), in sharp contrast with (3.11) above.

More generally, let L_{p}(N , φ), or briefly L_{p}(N , φ), be another non-commutative (semi-finite) L_{p}-space. Consider f_{k} ∈ B ⊗ L_{p}(φ) and let

F = ∑ f_{k} ⊗ x_{k} ∈ B ⊗ L_{p}(φ × τ)

where {x_{k}} is as in Theorem 11.2. We have then:

Theorem 11.8. Let p = 2n and let C = C_{ψ,p} be the constant appearing in (11.4). Then for any F as above we have

∥F ∥_{(p)} ≤ C max {∥∑ f_{k} ˙⊗f_{k}^{∗}∥_{(p/2)}^{1/2} , ∥∑ f_{k}^{∗} ˙⊗f_{k}∥_{(p/2)}^{1/2}} .  (11.7)

Proof. Repeating the steps of the proof of Theorem 11.2, all we need to do is majorize

∥φ̂ ( ∑_{(k_{1},...,k_{2n})∼ν} f_{k_{1}} ˙⊗f_{k_{2}}^{∗} ˙⊗ ⋯ ˙⊗f_{k_{2n}}^{∗} )∥

by the right side of (11.7). This is established in Lemma 11.11 below that is a rather easy adaptation to our Λ_{p}-setting of [6, Lemma 2].

By the same argument as in Remark 11.4, the case of the free generators of the free group can be deduced from the “free-Gaussian” one. Indeed, let (c_{k}) be a free circular family (sometimes called “complex free-Gaussian”). The polar decomposition c_{k} = u_{k}|c_{k}|, is such that the ∗-distribution of (u_{k}) is identical to that of a free family of Haar unitaries in the sense of [35], or equivalently (u_{k}) has the same ∗-distribution as that of the free generators λ(g_{k}) in the von Neumann algebra of the free group with infinitely many generators. Moreover, a simple calculation relative to the circular distribution yields ∥c_{k}∥_{1} = 8/3π. These observations lead us to :

Corollary 11.9. With the same notation as in Corollary 11.6, let F̃ = ∑ f_{k}⊗ λ(g_{k}). We have then

(3π/4)^{−1}∥F̃∥_{(p)} ≤ max {∥∑ f_{k} ˙⊗f_{k}^{∗}∥_{(p/2)}^{1/2} , ∥∑ f_{k}^{∗} ˙⊗f_{k}∥_{(p/2)}^{1/2}} ≤ ∥F̃∥_{(p)}

(11.8)

Proof. Let τ̃ denote the normalized trace on the von Neumann algebra of the free group with generators (g_{k}). Let E denote the conditional expectation equal to the orthogonal projection from L_{2}(φ ⊗ τ̃) onto L_{2}(φ) ⊗ 1. Then E(F̃ ˙⊗F̃^{∗}) = ∑ f_{k}^{∗} ˙⊗f_{k}. Since ∥F̃∥_{(p)}^{2} = ∥F̃ ˙⊗F̃^{∗}∥_{p/2} and ∥F̃ ˙⊗F̃^{∗}∥_{p/2} ≥ ∥E(F̃ ˙⊗F̃^{∗})∥_{p/2} by (8.13), the right hand side follows. To prove the left hand side, consider F = ∑ f_{k} ⊗ c_{k} = ∑ f_{k} ⊗ u_{k}|c_{k}| with (c_{k}) free circular as above and note that by the preceding observations (this is similar to Remark 11.4) we have ∑ f_{k} ⊗ u_{k} = (3π/8)(Id ⊗ E_{1})(∑ f_{k} ⊗ u_{k}|c_{k}|) where E_{1} denotes the conditional expectation from the von Neumann algebra generated by {c_{k}} onto the one generated by {u_{k}}. Since {u_{k}} and {λ(g_{k})} have identical ∗-moments, we find

∥F̃∥_{(p)} = ∥ ∑ f_{k}⊗u_{k}∥_{(p)} ≤ (3π/8)∥ ∑ f_{k}⊗u_{k}|c_{k}|∥_{(p)} = (3π/8)∥ ∑ f_{k}⊗c_{k}∥_{(p)}

and hence the left hand side of (11.8) follows from (11.7), recalling that C ≤ 2 when (x_{k}) is a free circular sequence.

Remark 11.10. A more careful estimate probably yields the preceding Corollary with the constant 2 in place of 3π/4.

Lemma 11.11. With the preceding notation let

S(ν) = ∑_{(k_{1},...,k_{2n})∼ν} f_{k_{1}} ˙⊗f_{k_{2}}^{∗} ˙⊗ ⋯ ˙⊗f_{k_{2n}}^{∗},

and let ν′_{0} (resp. ν′′_{0}) denote the partition of [1, …, 2n] into consecutive pairs of the form {1, 2}, {3, 4}, … (resp. {2n, 1}, {2, 3}, {4, 5}, …). We have then

∥φ̂(S(ν))∥ ≤ max{∥φ̂(S(ν′_{0}))∥, ∥φ̂(S(ν′′_{0}))∥}.

Sketch of Proof. We set

C = sup ∥φ̂(S(ν))∥

where the sup runs over all pair partitions ν in P_{2}(2n). By the cyclicity of the trace (see (8.1)) we may assume that k_{1}, …, k_{2n} is such that for some j with n < j ≤ 2n, the pair {k_{n}, k_{j}} is a block of our partition ν. Let

F (ν) = ∑_{k_{1},...,k_{2n}∼ν} f_{k_{1}} ˙⊗f_{k_{2}}^{∗} ˙⊗ ⋯ ˙⊗f_{k_{2n}}^{∗}.

We may rewrite F (ν) as

F (ν) = ∑_{α} ∑_{β} a_{α,β} ˙⊗b_{α,β}  (11.9)

where α represents the set of indices k_{j} such that the pair containing j is split by the partition [1, …, n][n + 1, …, 2n], and β represents the remaining indices, and the sum is restricted to (k_{1}, …, k_{2n}) ∼ ν. Since the indices in β correspond to pairs of indices {k_{i}, k_{j}} with {i, j} included either in [1, …, n] or in [n + 1, …, 2n], we can rewrite the sum (11.9) as

F (ν) = ∑_{α} ∑_{β′,β′′} α_{α,β′} ˙⊗b_{α,β′′}.

Then

S(ν) = ∑_{α} x_{α} ˙⊗y_{α}

with x_{α} = ∑_{β′} a_{α,β′} and y_{α} = ∑_{β′′} b_{α,β′′}. By (8.2) we find

∥φ̂(S(ν))∥ ≤ ∥φ̂ (∑_{α} x_{α} ˙⊗x_{α}^{∗})∥^{1/2} ∥φ̂ (∑_{α} y_{α}^{∗} ˙⊗y_{α})∥^{1/2} .

But now ∑_{α} x_{α} ⊗ x_{α}^{∗} is a sum of the kind S(ν′) for some ν′ but for which we know (by our initial choice relative to the pair {n, j}) that the pair {n, n + 1} appears in ν′. If we then iterate the argument in the style of [6] we end up with a number 0 < θ < 1 such that we have either

∥φ̂(S(ν))∥ ≤ (C′)^{θ}C^{1−θ}

or

∥φ̂(S(ν))∥ ≤ (C′′)^{θ}C^{1−θ}

where C′ = ∥φ̂(S(ν′_{0}))∥ and C′′ = ∥φ̂(S(ν′′_{0}))∥. Thus we conclude that

C ≤ (max(C′, C′′))^{θ}C^{1−θ}

and hence C ≤ max(C′, C′′). Since S(ν′_{0}) = (∑ f_{k} ˙⊗f_{k}^{∗})^{˙⊗n} and S(ν′′_{0}) = (∑ f_{k}^{∗} ˙⊗f_{k})^{˙⊗n}, this completes the proof.

By a spin system we mean a system of anticommuting self-adjoint unitaries assumed realized over a non-commutative probability space (M, τ). In the q-Gaussian case with q = −1, Theorem 11.2 describes the closed span of a spin system in Λ_{p}, and exactly for the same reason as in [7] we obtain optimal constants for those.

Corollary 11.12. If (x_{k}) is a spin system, then (11.4) holds with the same optimal constant C_{ψ,p} as in the Gaussian case. In particular, this constant grows like \sqrt{p} when p → ∞. Moreover, the same result holds for the (Rademacher) sequence (ε_{k}), and again the Gaussian constant is optimal.

Proof. Let ψ(q) denote the function ψ for a q-Gaussian system. By Bożejko and Speicher’s results (see [7]) we have ψ(q)(ν) = q^{i(ν)} where i(ν) is the crossing number of the the partition ν. This implies |ψ(q)(ν)| = ψ(|q|)(ν) and hence also C_{ψ(q),p} = C_{ψ(|q|),p}. In particular, any spin system (x_{k}) satisfies (11.4) with the constant C_{ψ(1),p}, i.e. the same constant as in the Gaussian case. We now address the Rademacher case. Just as in [7] we use the fact that the sequences (x_{k} ⊗ x_{k}) and (ε_{k}) have the same distribution. We then apply (11.7) to ∑ f_{k} ⊗ x_{k} with f_{k} = b_{k} ⊗ x_{k}. Recalling that the x_{k}’s are unitary, we find ∑ f_{k}^{∗} ˙⊗f_{k} = ∑ b̄_{k} ⊗ b_{k} ⊗ 1 and ∑ f_{k} ˙⊗f_{k}^{∗} = ∑ b_{k} ⊗ b̄_{k} ⊗ 1. This gives us ∥ ∑ b_{k} ⊗ε_{k}∥_{(p)} ≤ C∥ ∑ b_{k} ⊗b̄_{k}∥^{1/2} where C = C_{ψ(1),p} is the Gaussian constant. By the central limit theorem, the latter is optimal.

In the rest of this section we turn to the span of an i.i.d. sequence of Gaussian random matrices of size N × N in Λ_{p}. We will use ideas from [12] and [6]. We analyze the dependence in N using a concentration of measure argument. Let {g_{ij} | i, j ≥ 1} be a doubly indexed family of complex valued Gaussian random variables such that 𝔼g_{ij} = 0 and 𝔼|g_{ij}|^{2} = 1. Let Y^{(N)} be the random N × N matrix defined by

Y^{(N)}(i, j) = N^{−1/2}g_{ij}.

Let Y_{1}^{(N)}, Y_{2}^{(N)}, … be an i.i.d. sequence of copies of Y^{(N)} on some probability space (Ω, A, ℙ). We will view (Y_{j}^{(N)})_{j≥1} as a sequence in L_{p}(ℙ × τ_{N}) where τ_{N} denotes the normalized trace on M_{N}. By the Appendix §15, we know that, for any even p ≥ 2, (Y_{j}^{(N)})_{j≥1} has p-th moments defined by pairings via the function

Y^{(N)}(ν) = 𝔼τ_{N}(Y^{(N)ν})

where Y^{(N)ν} = Y_{k_{1}}^{(N)}Y_{k_{2}}^{(N)^{∗}} … Y_{k_{p−1}}^{(N)}Y_{k_{p}}^{(N)^{∗}} for k = (k_{j}) such that k_{i} = k_{j} if and only if (i, j) belong to the same block of ν. It is easy to see that the distribution of Y^{(N)ν} does not depend on the choice of such a k. Moreover, ψ^{(N)}(ν) ≥ 0 for any ν since ψ^{(N)}(ν) is a sum of terms of the form

𝔼 (Y_{k_{1}}^{(N)}(i_{1}, j_{1})\overline{Y_{k_{2}}^{(N)}(i_{2}, j_{2})} … \overline{Y_{k_{p}}^{(N)}(i_{p}, j_{p})})

and, when k ∼ ν, these are either = 0 or = N^{−p/2}(𝔼|g_{11}|^{2})^{p/2} = N^{−p/2}. Therefore we again have

∑ |ψ^{(N)}(ν)| = ∑ ψ^{(N)}(ν) = 𝔼τ_{N}(|Y^{(N)}|^{p}).

By (11.4) we have:

Corollary 11.13. Let p = 2n and let (b_{k}) be any finite sequence in B = B(H). Let f ∈ ∑ b_{k} ⊗ Y_{k}^{(N)} ∈ B ⊗ L_{p}(ℙ × τ_{N}). We have

∥f∥_{(p)} ≤ (𝔼τ_{N}(|Y^{(N)}|^{p}))^{1/p} ∥∑ b_{k} ⊗ b̄_{k}∥^{1/2}

and this constant is optimal.

12 Non-commutative martingale inequalities

In this section, we assume given a filtration M_{0} ⊂ M_{1} ⊂ ⋯ of von Neumann subalgebras of M. We assume for simplicity that M coincides with the von Neumann algebra generated by ∪M_{n}. We will denote again by 𝔼_{n} the conditional expectation with respect to M_{n}. Then to any f in L_{p}(τ) (1 ≤ p < ∞) we can associate a martingale (f_{n}) (defined by f_{n} = 𝔼_{n}(f)) that converges in L_{p}(τ) to f. We will continue to denote d_{0} = f_{0} and d_{n} = f_{n} − f_{n−1}. It is natural to expect that Theorems 4.1 and 5.5 will extend to the noncommutative case. However, at the time of this writing, we have completed this task only for p = 4. We also proved below (see Theorem 13.1) a one sided version of (4.1) using the notion of p-orthogonal sums. Let H_{1}, H_{2} be two Hilbert spaces. To lighten the notation in the rest of this section we set B_{1} = B(H_{1}) and B_{2} = B(H_{2}). It is useful to observe that for any f_{1} ∈ B_{1} ⊗ L_{4}(τ) and f_{2} ∈ B_{2} ⊗ L_{4}(τ) we have

τ̂(f_{1}^{∗} ˙⊗f_{1} ˙⊗f_{2}^{∗} ˙⊗f_{2}) ≈ τ̂(f_{1} ˙⊗f_{2}^{∗} ˙⊗f_{2} ˙⊗f_{1}^{∗}) ≻ 0.  (12.1)

Indeed, the first sign ≈ is by the trace property while sign ≻ 0 holds because f_{1} ˙⊗f_{2}^{∗} ˙⊗f_{2} ˙⊗f_{1}^{∗} ≈ F ˙⊗F^{∗} with F = f_{1} ˙⊗f_{2}^{∗} ∈ B_{1} ⊗ \overline{B}_{2} ⊗ L_{2}(τ). In the next lemma, we extend this observation to τ̂(f_{1}^{∗} ˙⊗f_{1} ˙⊗T (f_{2}^{∗} ˙⊗f_{2})) where T: L_{2}(τ) → L_{2}(τ) is a completely positive map (e.g. a conditional expectation). The reader can convince himself easily that the simplest case of maps of the form T (x) = ∑ a_{k}^{∗}xa_{k} (a_{k} ∈ M), follows immediately from (12.1).

Lemma 12.1. With the preceding notation, let B = B_{1} ⊗ \overline{B}_{2} = B(H_{1} ⊗_{2} \overline{H}_{2}). For any completely positive map T: L_{2}(τ) → L_{2}(τ), we have for any f_{1} ∈ B_{1} ⊗ L_{4}(τ) and f_{2} ∈ B_{2} ⊗ L_{4}(τ)

τ̂(f_{1}^{∗} ˙⊗f_{1} ˙⊗T (f_{2}^{∗} ˙⊗f_{2})) ≻ 0

where the latter element is identified with an element of B ⊗ \overline{B}, via the permutation (1 2 3 4) → (2 3 1 4) of the tensorial factors that takes \overline{B}_{1} ⊗ B_{1} ⊗ \overline{B}_{2} ⊗ B_{2} to B ⊗ \overline{B}.

Remark 12.2. Let E = L_{4}(τ) ⊗ \overline{L_{4}(τ)}. Let T: L_{2}(τ) → L_{2}(τ) be a completely positive map, so that for any finite sequence a_{1}, …, a_{n} in L_{4}(τ) the matrix [T (a_{i}^{∗}a_{j})] is in L_{2}(M_{n}(M))_{+}. Let Φ: E ⊗ \overline{E} → ℂ be the bilinear form defined by

Φ(a_{1} ⊗ b̄_{1} ⊗ \overline{a_{2} ⊗ b̄_{2}}) = τ(a_{2}^{∗}a_{1}T (b_{1}^{∗}b_{2})).

We claim that Φ is positive definite on E ⊗ \overline{E}, i.e. Φ(e ⊗ ē) ≥ 0 for any e in E. Indeed, if e = ∑ a_{i} ⊗ b̄_{i} ∈ E we have

Φ(e ⊗ ē) = ∑_{ij} τ(a_{j}^{∗}a_{i}T (b_{i}^{∗}b_{j})),

so that, if τ_{n} denote the trace on M_{n}(M), we have Φ(e ⊗ ē) = τ_{n}(αβ) = τ_{n}(a^{1/2}βα^{1/2}) ≥ 0 where β_{ij} = T (b_{i}^{∗}b_{j}) α_{ij} = a_{i}^{∗}a_{j} and of course α ≥ 0 and β ≥ 0. This proves our claim.

Proof of Lemma 12.1. Let L_{p} = L_{p}(τ). Consider f_{1} ⊗ f̄_{2} ∈ B_{1} ⊗ L_{4} ⊗ \overline{B}_{2} ⊗ L̄_{4}. Let g ∈ B ⊗ E be the element obtained by the natural permutation of factors from f_{1} ⊗ f̄_{2}. Then an easy verification shows that

τ̂(f_{1}^{∗} ˙⊗f_{1} ˙⊗T (f_{2}^{∗} ˙⊗f_{2})) ≈ (I ⊗ Φ)(g ⊗ ḡ)

or more precisely with the notation indicated in Lemma 8.5

τ̂(f_{1}^{∗} ˙⊗f_{1} ˙⊗T (f_{2}^{∗} ˙⊗f_{2})) ≈ (Φ)_{24}(g ⊗ ḡ)

so that Lemma 12.1 follows from Lemma 8.5 and the preceding Remark.

Proposition 12.3. Let (f_{n})_{n≥0} be a martingale in B ⊗ L_{4}(τ). Assume for simplicity f = f_{N} for some N ≥ 0. Let g = f^{∗} ˙⊗f − ∑ d_{n}^{∗} ˙⊗d_{n}. We have then

∥g∥_{(2)} ≤ ∥f∥_{(4)}(∥σ_{r}∥_{(2)}^{1/2} + ∥σ_{c}∥_{(2)}^{1/2}).  (12.2)

where

σ_{r} = ∑ 𝔼_{n−1}(d_{n} ˙⊗d_{n}^{∗}) and σ_{c} = ∑ 𝔼_{n−1}(d_{n}^{∗} ˙⊗d_{n}).

Proof. As usual we start by g = x + y with x = ∑ d_{n}^{∗} ˙⊗f_{n−1} and y = ∑ f_{n−1}^{∗} ˙⊗d_{n}, so that ∥g∥_{(2)} ≤ ∥x∥_{(2)} + ∥y∥_{(2)}. Then

∥x∥_{(2)}^{2} = ∥τ̂(x ˙⊗x^{∗})∥ = ∥τ̂ (∑ d_{n}^{∗} ˙⊗f_{n−1} ˙⊗f_{n−1}^{∗} ˙⊗d_{n})∥ .

Let δ_{n} = f − f_{n−1}. Note that since 𝔼_{n−1}(δ_{n}) = 0

𝔼_{n−1}(f ˙⊗f^{∗}) = f_{n−1} ˙⊗f_{n−1}^{∗} + 𝔼_{n−1}(δ_{n} ˙⊗δ_{n}^{∗})

and hence

τ̂ (∑ d_{n}^{∗} ˙⊗f_{n−1} ˙⊗f_{n−1}^{∗} ˙⊗d_{n}) =

= τ̂ (∑ d_{n}^{∗} ˙⊗𝔼_{n−1}(f ˙⊗f^{∗}) ˙⊗d_{n}) − τ̂ (∑ d_{n}^{∗} ˙⊗𝔼_{n−1}(δ_{n} ˙⊗δ_{n}^{∗}) ˙⊗d_{n}) ,

By the trace property and by Lemma 12.1, these last three terms can all be viewed as ≻ 0 in a suitable permutation of the factors. This shows by (2.6)

∥τ̂ (∑ d_{n}^{∗} ˙⊗f_{n−1} ˙⊗f_{n−1}^{∗} ˙⊗d_{n})∥ ≤ ∥τ̂ (∑ d_{n}^{∗} ˙⊗𝔼_{n−1}(f ˙⊗f^{∗}) ˙⊗d_{n})∥ .

Since

τ̂ (∑ d_{n}^{∗} ˙⊗𝔼_{n−1}(f ˙⊗f^{∗}) ˙⊗d_{n}) ≈ τ̂ (𝔼_{n−1}(f ˙⊗f^{∗}) ˙⊗ ∑ d_{n} ˙⊗d_{n}^{∗}) ,

and since 𝔼_{n−1} is self-adjoint, we have

∥τ̂ (∑ d_{n}^{∗} ˙⊗𝔼_{n−1}(f ˙⊗f^{∗}) ˙⊗d_{n})∥ = ∥τ̂ (f ˙⊗f^{∗} ˙⊗ ∑ 𝔼_{n−1}(d_{n} ˙⊗d_{n}^{∗}))∥ ,

thus we find

∥x∥_{(2)}^{2} ≤ ∥τ̂(f ˙⊗f^{∗} ˙⊗σ_{r})∥ ≤ ∥f∥_{(4)}^{2}∥σ_{r}∥_{(2)}.

A similar reasoning leads to

∥y∥_{(2)}^{2} ≤ ∥f∥_{(4)}^{2}∥σ_{c}∥_{(2)},

so we conclude

∥g∥_{(2)} ≤ ∥f∥_{(4)}(∥σ_{r}∥_{(2)}^{1/2} + ∥σ_{c}∥_{(2)}^{1/2}).

To complete the case p = 4, we need to check the non-commutative extension of Lemma 4.2 as follows:

Lemma 12.4. Let θ_{n} be any finite sequence in B ⊗ L_{4}(τ), let β_{n} = θ_{n}^{∗} ˙⊗θ_{n} and α_{n} = 𝔼_{n}(β_{n}). Then

∥∑ α_{n}∥_{(2)} ≤ 2 ∥∑ β_{n}∥_{(2)} .

Proof. The proof is essentially the same as for Lemma 4.2. We just need to observe that if n ≤ k we have

τ̂(α_{n} ˙⊗α_{k}) = τ̂(α_{n} ˙⊗β_{k}),

but also by Lemma 12.1 (and the trace property)

τ̂(α_{n} ˙⊗β_{k}) ≈ τ̂(β_{k} ˙⊗α_{n}) ≻ 0,  ∀n, k

so that again we have

∥∑_{n≤k} τ̂(α_{n} ˙⊗β_{k})∥ ≤ ∥∑_{n,k} τ̂(α_{n} ˙⊗β_{k})∥ = ∥τ̂(α ˙⊗β)∥ ≤ ∥α∥_{(2)}∥β∥_{(2)}

and similarly for ∥ ∑_{n>k} ∥.

Let S_{r} = ∑ d_{j} ˙⊗d_{j}^{∗} abd S_{c} = ∑ d_{j}^{∗} ˙⊗d_{j}. Applying this Lemma to Proposition 12.3, we find

∥g∥_{(2)} ≤ 2\sqrt{2}∥f∥_{(4)} max\{∥S_{r}∥_{(2)}^{1/2}, ∥S_{c}∥_{(2)}^{1/2}\}.

We then obtain by the same reasoning as for the commutative case:

Corollary 12.5. There is a constant C such that for any finite martingale f_{0}, …, f_{N} in L_{4}(τ) we have

C^{−1} max {∥∑ d_{j} ˙⊗d_{j}^{∗}∥_{(2)}^{1/2} , ∥∑ d_{j}^{∗} ˙⊗d_{j}∥_{(2)}^{1/2}} ≤ ∥f∥_{(4)} ≤

≤ C max {∥∑ d_{j} ˙⊗d_{j}^{∗}∥_{(2)}^{1/2} , ∥∑ d_{j}^{∗} ˙⊗d_{j}∥_{(2)}^{1/2}} .

Remark 12.6. We leave as an open problem whether the extension of the left hand side of Corollary 12.5 is valid for any even integer p > 4. Note however that the right hand side is proved below as a consequence of Theorem 13.1.

We will now extend Theorem 5.5 to the non-commutative case for p = 4. Given f ∈ B ⊗ L_{4}(τ) let us denote

∥f∥_{[4]} = max{∥τ̂(∑ d_{n} ˙⊗d_{n}^{∗} ˙⊗d_{n} ˙⊗d_{n}^{∗})∥^{1/4}, ∥σ_{r}∥_{(2)}^{1/2}, ∥σ_{c}∥_{(2)}^{1/2}}.

Corollary 12.7. For any finite martingale f_{0}, …, f_{N} in L_{4}(τ) we have

2C^{−1}∥f∥_{[4]} ≤ ∥f∥_{(4)} ≤ 2C∥f∥_{[4]},  (12.3)

where C is as in the preceding statement.

Proof. By the preceding Corollary and by Lemma 12.4 we have max{∥σ_{r}∥_{(2)}^{1/2}, ∥σ_{c}∥_{(2)}^{1/2}} ≤ 2C∥f∥_{(4)}. Moreover, by (12.1) we have τ̂(∑ d_{n} ˙⊗d_{n}^{∗} ˙⊗d_{n} ˙⊗d_{n}^{∗}) ≺ τ̂(∑ d_{n} ˙⊗d_{n}^{∗}) ˙⊗(∑ d_{j} ˙⊗d_{j}^{∗})^{∗}), and hence

∥τ̂(∑ d_{n} ˙⊗d_{n}^{∗} ˙⊗d_{n} ˙⊗d_{n}^{∗})∥^{1/4} ≤ ∥∑ d_{j} ˙⊗d_{j}^{∗}∥_{(2)}^{1/2}. Therefore, the left hand side of (12.3) follows. To prove the right hand side, we will use the preceding Corollary. Let x_{n} = d_{n}^{∗} ˙⊗d_{n} and y_{n} = d_{n} ˙⊗d_{n}^{∗}. We have

∑ x_{n} = σ_{c} + ∑ δ_{n}

where δ_{n} = x_{n} − 𝔼_{n−1}x_{n}. Then, by the triangle inequality ∥ ∑ x_{n}∥_{(2)} ≤ ∥σ_{c}∥_{(2)} +∥ ∑ δ_{n}∥_{(2)} and by the orthogonality of the martingale differences (δ_{n}) we have ∥ ∑ δ_{n}∥_{(2)}^{2} = ∥ ∑ τ̂(δ_{n}^{∗} ˙⊗δ_{n})∥. But by Remark 8.9 we have τ̂(δ_{n}^{∗} ˙⊗δ_{n}) ≺ τ̂(x_{n}^{∗} ˙⊗x_{n}) and hence also ∑ τ̂(δ_{n}^{∗} ˙⊗δ_{n}) ≺ ∑ τ̂(x_{n}^{∗} ˙⊗x_{n}) from which follows by Lemma 2.3 that ∥ ∑ τ̂(δ_{n}^{∗} ˙⊗δ_{n})∥ ≤ ∥ ∑ τ̂(x_{n}^{∗} ˙⊗x_{n})∥. Recapitulating, we find ∥ ∑ x_{n}∥_{(2)} ≤ ∥σ_{c}∥_{(2)}+∥ ∑ τ̂(x_{n}^{∗} ˙⊗x_{n})∥^{1/2}, and a fortiori

∥ ∑ x_{n}∥_{(2)}^{1/2} ≤ ∥σ_{c}∥_{(2)}^{1/2} + ∥ ∑ τ̂(x_{n}^{∗} ˙⊗x_{n})∥^{1/4} ≤ 2∥f∥_{[4]}.

Since a similar argument applies to majorize ∥ ∑ y_{n}∥_{(2)}, by Corollary 12.5 we obtain

C^{−1}∥f∥_{(4)} ≤ max{∥ ∑ x_{n}∥_{(2)}^{1/2}, ∥ ∑ y_{n}∥_{(2)}^{1/2}} ≤ 2∥f∥_{[4]}.

13 p-orthogonal sums

Let L_{p}(τ) be as before the “non-commutative” L_{p}-space associated to a von Neumann algebra equipped with a standard (= faithful, normal) semi-finite trace. (Of course, if M is commutative, we recover the classical L_{p} associated to a measure space.) Let p ≥ 2 be an even integer. A family d = (d_{i})_{i∈I} is called p-orthogonal if, for any injective function g: [1, 2, …, p] → I we have

τ(d_{g(1)}^{∗}d_{g(2)}d_{g(3)}^{∗}d_{g(4)} … d_{g(p−1)}^{∗}d_{g(p)}) = 0.

Clearly, any martingale difference sequence is p-orthogonal, but the class of p-orthogonal sums is more general. In the commutative case, i.e. for classical random variables, this notion is very close to that of “multiplicative sequence” already considered in the literature, see the references in [24], on which this section is modeled. By a natural extension, we will say that a sequence (d_{j})_{j∈I} in B ⊗ L_{p}(τ) is p-orthogonal if for any injective function g: [1, 2, …, p] → I as before we have

τ̂(d_{g(1)}^{∗} ˙⊗d_{g(2)} ˙⊗ … ˙⊗d_{g(p−1)}^{∗} ˙⊗d_{g(p)}) = 0.

The method used in [24], that is based on a combinatorial formula involving the “Möbius function”, is particularly easy to adapt to our setting where Λ_{p} takes the place of L_{p}. We will use crucially some well known ideas from the combinatorial theory of partitions, which can be found, for instance, in the book [1]. We denote by P_{n} the lattice of all partitions of [1, …, n], equipped with the following order: we write σ ≤ π (or equivalently π ≥ σ) when every “block” of the partition σ is contained in some block of π. Let ˙0 and ˙1 be respectively the minimal and maximal elements in P_{n}, so that ˙0 is the partition into n singletons and ˙1 the partition formed of the single set {1, …, n}. We denote by ν(π) the number of blocks of π (so that ν(˙0) = n and ν(˙1) = 1). For any π in P_{n} and any i = 1, 2, …, n, we denote by r_{i}(π) the number of blocks (possibly = 0) of π of cardinality i. In particular, we have ∑_{1}^{n} ir_{i}(π) = n and ∑_{1}^{n} r_{i}(π) = ν(π). Given two partitions σ, π in P_{n} with σ ≤ π we denote by μ(σ, π) the Möbius function, which has the following fundamental property: Let V be a vector space. Consider two functions Φ: P_{n} → V and Ψ: P_{n} → V. If Ψ(σ) = ∑_{π≤σ} Φ(π), then Φ(σ) = ∑_{π≤σ} μ(π, σ)Ψ(π).

Essentially equivalently, if Ψ(σ) = ∑_{π≥σ} Φ(π), then Φ(σ) = ∑_{π≥σ} μ(σ, π)Ψ(π). In particular we have:

∀ σ ≠ ˙0 ∑_{˙0≤π≤σ} μ(π, σ) = 0.  (13.1)

The last assertion follows from the above fundamental property applied with Φ equal to the delta function at ˙0 (i.e. Φ(π) = 0 ∀ π ≠ ˙0 and Φ(˙0) = 1) and Ψ ≡ 1. We also recall Schützenberger’s theorem (see [1]):

For any π we have

μ(˙0, π) = ∏_{i=1}^{n}[(−1)^{i−1}(i − 1)!]^{r_{i}(π)},

and consequently

∑_{π∈P_{n}} |μ(˙0, π)| = n!.

We now apply these results to set the stage for the questions of interest to us. Let E_{1}, …, E_{n}, V be vector spaces equipped with a multilinear form (= a “product”)

φ: E_{1} × ⋯ × E_{n} → V.

Let I be a finite set. For each k = 1, 2, …, n and i ∈ I, we give ourselves elements d_{i}(k) ∈ E_{k}, and we form the sum

F_{k} = ∑_{i∈I} d_{i}(k).

Then we are interested in “computing” the quantity φ(F_{1}, …, F_{n}). We have obviously

φ(F_{1}, …, F_{n}) = ∑_{g} φ(d_{g(1)}(1), …, d_{g(n)}(n))

where the sum runs over all functions g: [1, 2, …, n] → I. Let π(g) be the partition associated to g, namely the partition obtained from ⋃_{i∈I} g^{−1}({i}) after

deletion of all the empty blocks. We can write

φ(F_{1}, …, F_{n}) = ∑_{σ∈P_{n}} Φ(σ)

where Φ(σ) = ∑_{g: π(g)=σ} φ(d_{g(1)}(1), …, d_{g(n)}(n)). Let Ψ(σ) = ∑_{π≥σ} Φ(π).

Using (13.1) (with σ, π exchanged), we have then:

φ(F_{1}, …, F_{n}) = Φ(˙0) + ∑_{˙0<σ} Φ(σ) = Φ(˙0) + ∑_{˙0<σ} ∑_{π≥σ} μ(σ, π)Ψ(π)  (13.2)

= Φ(˙0) + ∑_{˙0<π} Ψ(π) · ∑_{˙0<σ≤π} μ(σ, π) = Φ(˙0) − ∑_{˙0<π} Ψ(π)μ(˙0, π). (13.3)

Recapitulating, we found:

φ(F_{1}, …, F_{n}) = Φ(˙0) − ∑_{˙0<π} Ψ(π)μ(˙0, π)  (13.4)

where

Φ(˙0) = ∑_{g injective} φ(d_{g(1)}(1), …, d_{g(n)}(n))

and

Ψ(π) = ∑_{g: π(g)≥π} φ(d_{g(1)}(1), …, d_{g(n)}(n)).

Theorem 13.1. Let p = 2n be an even integer > 2. Then for any p-orthogonal finite sequence (d_{j})_{j∈I} in B ⊗ L_{p}(τ) we have

∥∑ d_{j}∥_{(p)} ≤ (3π/2)p max {∥∑ d_{j} ˙⊗d_{j}^{∗}∥_{(p/2)}^{1/2} , ∥∑ d_{j}^{∗} ˙⊗d_{j}∥_{(p/2)}^{1/2}} . (13.5)

Proof. This proof is modeled on that in [24] so we will be deliberately sketchy. Let f = ∑ d_{j}. We can write

τ̂[(f^{∗} ˙⊗f)^{˙⊗n}] = − ∑_{˙0<π} μ(˙0, π)Ψ(π)

where Φ and Ψ are defined by

Φ(σ) = ∑_{g: π(g)=σ} τ̂(d_{g(1)}^{∗} ˙⊗d_{g(2)} … ˙⊗d_{g(p−1)}^{∗} ˙⊗d_{g(p)})

and Ψ = ∑_{σ≥π} Φ(σ), or equivalently,

Ψ(π) = ∑_{g∼σ} τ̂(d_{g(1)}^{∗} ˙⊗ ⋯ ˙⊗d_{g(p)})

where (as in §11) g ∼ σ means that g(i) = g(j) whenever i, j are in the same block of σ. Here the functions Φ and Ψ take values in \overline{B} ⊗ B ⊗ ⋯ ⊗ \overline{B} ⊗ B where \overline{B} ⊗ B is repeated n-times . Let α = 3π/4 as in [24]. Arguing as in [24, p. 912] we see that it suffices to prove that

∥Ψ(π)∥ ≤ (αΔ)^{p−r_{1}(π)}∥f∥_{(p)}^{r_{1}(π)}  (13.6)

where Δ = max {∥∑ d_{j} ˙⊗d_{j}^{∗}∥_{(p/2)}^{1/2} , ∥∑ d_{j}^{∗} ˙⊗d_{j}∥_{(p/2)}^{1/2}}, and we recall that r_{1}(π) is the number of singletons in π. Let 𝔽_{I} be the free group with generators (g_{j})_{j∈I}, and let φ be the normalized trace on the von Neumann algebra of 𝔽_{I}. Let f_{k} = ∑_{i∈I} d_{i}(k) be a finite sum in B ⊗ L_{p}(τ), k = 1, …, p. We denote by

f̃_{k} = ∑_{i∈I} λ(g_{i}) ⊗ d_{i}(k)

the corresponding sum in L_{p}(φ × τ) ⊗ B. Note that by (11.8) we know that

∥f̃_{k}∥_{(p)} ≤ (3π/4) max {∥∑ d_{j} ˙⊗d_{j}^{∗}∥_{(p/2)}^{1/2} , ∥∑ d_{j}^{∗} ˙⊗d_{j}∥_{(p/2)}^{1/2}} .  (13.7)

Let π be a partition of [1, …, p]. Let B_{1} be the union of all the singletons in π and let B_{2} be the complement of B_{1} in [1, …, p]. By the construction in the proof of [24, Sublemma 3.3] for a suitable discrete group G there are F_{1}, …, F_{p} in L_{p}(τ_{G} ×τ)⊗B such that ∥F_{k}∥_{(p)} = ∥f̃_{k}∥_{(p)} for all k in B_{2}, ∥F_{k}∥_{(p)} = ∥f_{k}∥_{(p)} for all k in B_{1}, and also

τ̂ ( ∑_{π(g)≥π} d_{g(1)}(1) ˙⊗ ⋯ ˙⊗d_{g(p)}(p)) = { \atop (τ_{G} × τ)}[F_{1} ˙⊗ ⋯ ˙⊗F_{p}].

Then if we apply this to d_{j}(k) = d_{j}^{∗} if k is odd and d_{j}(k) = d_{j} if k is even we find by (8.4)

∥Ψ(π)∥ = ∥{ \atop (τ_{G} × τ)}(F_{1} ˙⊗ ⋯ ˙⊗F_{p})∥

≤ ∏_{k=1}^{p} ∥F_{k}∥_{(p)}

≤ ∥f∥_{(p)}^{|B_{1}|} · ∏_{k∈B_{2}} ∥f̃_{k}∥_{(p)}

and by (13.7) we obtain (13.6).

Corollary 13.2. Let p be an even integer. Assume τ(1) = 1. Let (f_{j}) be a p-orthogonal sequence in L_{p}(τ) that is orthonormal in L_{2}(τ). Consider a finite sequence (b_{j}) with b_{j} ∈ B. We have then

∥ ∑ b_{j} ⊗ b̄_{j}∥^{1/2} ≤ ∥∑ b_{j} ⊗ f_{j}∥_{(p)} ≤ (3π/2)p∥ ∑ b_{j} ⊗ b̄_{j}∥^{1/2}.

Let E_{p} denote the closed span of (f_{j}) in Λ_{p}(τ). Then E_{p} is completely isomorphic to OH, and moreover, the orthogonal projection P from L_{2}(τ) onto the span of (f_{j}) is c.b. on Λ_{p} with c.b. norm at most (3π/2)p.

Proof. The right hand side of the inequality follows from (13.5) since (d_{j}) = (b_{j} ⊗ f_{j}) is clearly p-orthogonal. By Corollary 8.4, the inclusion Λ_{p}(τ) → Λ_{2}(τ) = L_{2}(τ) has c.b norm 1. Using this the left hand side follows. This shows that E_{p} ≃ OH. The projection P can then be factorized as Λ_{p} → Λ_{2} → E_{2} → E_{p}, which implies ∥P: Λ_{p} → E_{p}∥_{cb} ≤ (3π/2)p, since the first two arrows are completely contractive.

14 Lacunary Fourier series in Λ_{p}

In this section, we review the results of [13] and [24] with Λ_{p} in place of L_{p}, and again we find the space OH appearing in place of R_{p} ∩ C_{p}. To save space, it will be convenient to adopt the general viewpoint in [24, §4], although this may seem obscure to a reader unfamiliar with [13].

Notation: Let 1 = ∑_{k∈J} P_{k} be an orthogonal decomposition of the identity of L_{2}(τ) on a semi-finite “non-commutative” measure space (M, τ). Let p = 2n be an even integer > 2. Let (d_{j})_{j∈I} be a finite family in B ⊗ L_{p}(τ). We set x^{ω} = x^{∗} if n is odd and x^{ω} = x in n is even.

Theorem 14.1. Let F be the set of all injective functions g: [1, 2, …, n] → I. For any g in F, we let x_{g} = d_{g(1)}^{∗} ˙⊗d_{g(2)} ˙⊗d_{g(3)}^{∗} … d_{g(n)}^{ω}. We define

N(d) = sup_{k∈J} card{g ∈ F | (P_{k} ⊗ I)(x_{g}) ≠ 0}.

Then

∥ ∑_{j∈I} d_{j}∥_{(p)}≤ [(4N(d))^{1/p} + p\frac{9π}{8} ] max \{∥∑ d_{j} ˙⊗d_{j}^{∗}∥_{(p/2)}^{1/2} , ∥∑ d_{j}^{∗} ˙⊗d_{j}∥_{(p/2)}^{1/2}\} .

Proof. Since we follow closely the ideas in [13] and [24] we will merely sketch the proofs. We have

∥f∥_{(p)}^{n} = ∥f^{∗} ˙⊗f ⋯ ˙⊗f^{ω}∥_{(2)}.

Arguing as in [24, p. 919] we find

f^{∗} ˙⊗f ⋯ ˙⊗f^{ω} = Φ(˙0) − ∑_{˙0<π∈P_{n}} μ(˙0, π)Ψ(π)

with Φ(σ) = ∑_{π(g)=σ} x_{g} and

Ψ(π) = ∑_{σ≤π} Φ(σ).

Note that Φ(˙0) = ∑_{g∈F} x_{g}. Using a suitable adaptation of [24, Sublemma 3.3] and replacing [24, (3.5)] by Corollary 11.9 above, we find:

∥Ψ(π)∥_{(2)} ≤ ∥f∥_{(p)}^{r_{1}(π)}(αΔ)^{n−r_{1}(π)}

where

Δ = max {∥∑ d_{j} ˙⊗d_{j}^{∗}∥_{(p/2)}^{1/2} , ∥∑ d_{j}^{∗} ˙⊗d_{j}∥_{(p/2)}^{1/2}} .

Let F_{k} = {g ∈ F | (id ⊗ P_{k})x_{g} ≠ 0} and Φ_{k} = (id ⊗ P_{k})Φ(˙0), so that

Φ(˙0) = ∑ Φ_{k} and Φ_{k} = ∑_{g∈F_{k}} x_{g}(k)

where x_{g}(k) = (id ⊗ P_{k})(x_{g}). By (2.3) we have by “orthogonality” of Φ_{k}

∥Φ(˙0)∥_{(2)}^{2} = ∥τ̂ (∑ Φ_{k} ˙⊗Φ_{k}^{∗})∥ .

Since card(F_{k}) ≤ N(d), by (2.8) and Lemma 8.6 we have

τ̂ (∑ Φ_{k} ˙⊗Φ_{k}^{∗}) ≺ N(d)τ̂ (∑_{k} ∑_{g∈F_{k}} x_{g}(k) ˙⊗x_{g}(k)^{∗}) =

= N(d)τ̂ (∑_{g∈F} x_{g} ˙⊗x_{g}^{∗}) .

Therefore, we find

\frac{1}{N(d)}∥Φ(˙0)∥_{(2)}^{2} ≤ ∥τ̂ (∑_{g∈F} x_{g} ˙⊗x_{g}^{∗})∥ ≤

≤ ∥τ̂ ( ∑_{g(1),...,g(n)} d_{g(1)}^{∗} ˙⊗ ⋯ ˙⊗d_{g(n)}^{ω} ˙⊗d_{g(n)}^{ω∗} … ˙⊗d_{g(1)})∥

and hence

∥Φ(˙0)∥_{(2)}^{2} ≤ N(d) ∥τ̂ ((∑ d_{j} ˙⊗d_{j}^{∗})^{˙⊗n})∥ = N(d) ∥∑ d_{j} ˙⊗d_{j}^{∗}∥_{(p/2)}^{p/2} .

Thus we may conclude by the same reasoning as in [24, p. 920].

We can now reformulate the main result of [13] with Λ_{p} in place of L_{p}:

Corollary 14.2. Fix an even integer p = 2n > 2. Let E ⊂ Γ be a subset of a discrete group Γ with unit e. For any γ in Γ let Z_{p}(γ, E) be the cardinality of the set of injective functions g: [1, …, n] → E such that

γ = g(1)g(2)^{−1}g(3) … g(n)^{w}

where g^{w} = g^{−1} if n is even and g^{w} = g if n is odd. We set

Z(E) = sup{Z_{p}(γ, E) | γ ∈ Γ}.

Then for any finitely supported family (b(t))_{t∈E} in B = B(H) we have

∥∑_{t∈E} λ(t) ⊗ b(t)∥_{(p)} ≤ ((4Z(E))^{1/p} + (9π/8)p) ∥∑ b(t) ⊗ \overline{b(t)}∥^{1/2} . (14.1)

Proof. Here L_{2}(τ) is the L_{2}-space associated to the usual trace on the von Neumann algebra associated to Γ. Any element in L_{2}(τ) has an orthonormal expansion in a “Fourier series” x = ∑_{t∈Γ} x(t)λ(t), so we can apply Theorem 14.1 to this orthogonal decomposition (with J = Γ). Note that if (t_{j}) are distinct elements in E and if d_{j} = λ(t_{j}) ⊗ b(t_{j}) we have N(d) ≤ Z(E). Lastly, we note that in the present situation, since L_{p}(φ) = ℂ, the term previously denoted by Δ coincides with

∥∑ b(t) ⊗ \overline{b(t)}∥^{1/2} .

We have also a (one sided) version of the Littlewood–Paley inequality for Λ_{p}:

Corollary 14.3. Consider a Fourier series of the form

f = ∑_{n>0} f̂(n)e^{int}

where n → f̂(n) is a finitely supported B(H)-valued function. Let

Δ_{n} = ∑_{2^{n}≤k<2^{n+1}} f̂(k)e^{ikt}

and let

S(f) = ∑ Δ_{n} ˙⊗\overline{Δ_{n}}.

There is an absolute constant C such that for any even integer p ≥ 2

∥f∥_{(p)} ≤ C_{p}∥S(f)∥_{(p/2)}^{1/2}.

Proof. It suffices to prove the inequality separately for the cases f = ∑_{m} Δ_{2m} and f = ∑_{m} Δ_{2m+1}. But then each of these cases follows from Theorem 14.1 and elementary arithmetic involving lacunary sequences.

It may be worthwhile to point out that in the commutative case, the following variant of Theorem 14.1 holds:

Theorem 14.4. Consider the same situation as in Theorem 14.1 but with M commutative so that L_{2}(τ) can be identified with L_{2}(Ω, μ). Let y_{g} = d_{g(1)} ˙⊗d_{g(2)} ˙⊗ ⋯ ˙⊗d_{g(n)} and let

N_{+}(d) = sup_{k∈J} card{g ∈ F | (P_{k} ⊗ I)(y_{g}) ≠ 0}.

Then

∥∑ d_{j}∥_{(p)} ≤ [(4N_{+}(d))^{1/p} + 9π/8] ∥∑ d_{j} ˙⊗d̄_{j}∥_{(p/2)}^{1/2} .

Proof. We argue exactly as for Theorem 14.1 except that we start instead from

∥f∥_{(p)}^{n} = ∥f ˙⊗ ⋯ ˙⊗f∥_{(2)}.

Remark 14.5. In particular, if E ⊂ Γ is a subset of a commutative group, let Z_{p+}(γ, E) be the cardinality of the set of injective g: [1, …, n] → E such that γ = g(1)g(2) … g(n) and let Z_{+}(E) = sup{Z_{p+}(γ, E) | γ ∈ Γ} < ∞. Then for any finitely supported family (b(t))_{t∈E} in B we have (14.1) with Z_{+}(E) in place of Z(E). Thus we obtain OH also for the span of certain Λ(p)-sets originally considered by Rudin [33], which are not Λ(p)_{cb}-sets in the sense of [13].

15 Appendix

The goal of this appendix is to clarify the relation between “moments defined by pairings” used in §9 (following [6]) and the well known Wick formula. The latter (probably going back independently to Ito and Wick) was used by Ito in connection with multiple Wiener integrals and Wiener chaos. Although we reformulate them using tensor products, the results below are all well known. We first consider the Gaussian case in a very general framework. Let B be a real vector space. Let X be a B-valued Gaussian variable. This means that for any ℝ-linear form ξ ∈ B^{∗}, the real valued variable ξ(X) is Gaussian with mean zero and variance equal to 𝔼ξ(X)^{2}. If B is a complex space, (e.g. if B = ℂ) we may view it a fortiori as a real one and the previous notion still makes sense. Let X = (X_{1}, …, X_{n}) be a Gaussian variable with values in B^{n}. Then X_{1} ⊗ ⋯ ⊗ X_{n} is a random variable with values in B^{⊗n}. When n is odd its mean vanishes. Let us assume that X = (X_{1}, …, X_{n}) is defined on (Ω, A, ℙ) and that n is even. Let π be a partition of [1, …, n] into K blocks. We will define a B^{⊗n}-valued random variable X^{⊗π} on (Ω, A, ℙ)^{⊗K} as follows: Assume that the blocks of π have been enumerated as α_{1}, …, α_{K}. We define ω̂_{j} for j = 1, …, n by setting ω̂_{j} = ω_{k} if j ∈ α_{k}. We then define

X^{⊗π}(ω_{1}, …, ω_{K}) = X_{1}(ω̂_{1}) ⊗ ⋯ ⊗ X_{n}(ω̂_{n}).

Note that the distribution (and hence all the moments) of X^{⊗π} do not depend on the particular enumeration (α_{1}, α_{2}, …) chosen to define it. In particular, 𝔼(X^{⊗π}) depends only on π. We now may state

Proposition 15.1. For any even integer n

𝔼(X_{1} ⊗ ⋯ ⊗ X_{n}) = ∑_{ν∈P_{2}(n)} 𝔼(X^{⊗ν}).  (15.1)

Proof. We will use the same trick as in [12, Prop. 1.5]) to deduce the formula from the rotational invariance of Gaussian distribution. Let X(s) = (X_{1}(s), …, X_{n}(s) be an i.i.d. sequence indexed by s ∈ ℕ of copies of X. By the invariance of Gaussian distributions, the variable X̂(s) = s^{−1/2}(X(1) + ⋯ + X(s)) has the same distribution as X. Therefore for any s, we have

𝔼(X_{1} ⊗ ⋯ ⊗ X_{n}) = 𝔼(X̂_{1}(s) ⊗ ⋯ ⊗ X̂_{n}(s))

and hence

𝔼(X_{1} ⊗ ⋯ ⊗ X_{n}) = lim_{s→∞} 𝔼(X̂_{1}(s) ⊗ ⋯ ⊗ X̂_{n}(s)).  (15.2)

Let E(s) = 𝔼(X̂_{1}(s) ⊗ ⋯ ⊗ X̂_{n}(s)). We have

E(s) = s^{−n/2} ∑_{g} 𝔼(X_{1}(g(1)) ⊗ ⋯ ⊗ X_{n}(g(n))), where the sum runs over all functions g: [1, …, n] → [1, …, s]. We claim that after elimination of all the (vanishing) odd terms and all the (asymptotically vanishing) terms such as

\frac{1}{s^{2}} ∑_{t≤s} X_{1}(t) ⊗ ⋯ ⊗ X_{4}(t)

we find

lim_{s→∞} E(s) = ∑_{ν∈P_{2}(n)} 𝔼(X^{⊗ν}).  (15.3)

Indeed, if we let

t(g) = 𝔼(X_{1}(g(1)) ⊗ ⋯ ⊗ X_{n}(g(n))

and if we denote by π(g) the partition of [1, …, n] defined by ⋃_{k≤s} g^{−1}(k) we have (eliminating vanishing terms)

E(s) = s^{−n/2} ∑_{g∈A_{s}} t(g)

where A_{s} is the set of g: [1, …, n] → [1, …, s] such that π(g) is a partition of [1, …, n] into blocks of even cardinality. For any such π, let

G_{s}(π) = {g ∈ A_{s} | π(g) = π}.

Let B_{s} ⊂ A_{s} be the set of all g’s such that π(g) is in P_{2}(n) (i.e. is a partition into pairs) so that

B_{s} = ⋃_{ν∈P_{2}(n)} G_{s}(ν).

Note that for any g in G_{s}(π) we have

t(g) = 𝔼(X^{⊗π}).

Let P′_{2}(n) denote the set of partitions π of [1, …, n] into blocks of even cardinality. We have

E(s) = s^{−n/2} ∑_{π∈P′_{2}(n)} |G_{s}(π)|𝔼(X^{⊗π}).

Note that P_{2}(n) ⊂ P′_{2}(n). Therefore

E(s) = ∑_{ν∈P_{2}(n)}s^{−n/2}|G_{s}(ν)|𝔼(X^{⊗ν})+∑_{π∈P′_{2}(n)\P_{2}(n)} s^{−n/2}|G_{s}(π)|𝔼(X^{⊗π}).

(15.4)

But now a simple counting argument shows that |G_{s}(ν)| = s(s − 1) … (s − \frac{n}{2} + 1) ≃ s^{n/2} and hence s^{−n/2}|G_{s}(ν)| → 1, while for any π in P′_{2}(n)\P_{2}(n) we have s^{−n/2}|G_{s}(π)| → 0. Thus taking the limit when s → ∞ in (15.4) yields our claim (15.3). By (15.2) this completes the proof.

To emphasize the connection with the classical Wick formula of which (15.1) is but an abstract form, let us state:

Corollary 15.2. Let X = (X_{j}) be a Gaussian sequence of real valued random variables (i.e. all their linear combinations are Gaussian). Then

𝔼(X_{1} ⋯ X_{n}) = ∑_{ν} ∏⟨X_{k_{j}}X_{ℓ_{j}}⟩

where the sum runs over all partitions ν of [1, …, n] into pairs, the product runs over all pairs {k_{j}, ℓ_{j}} (j = 1 ⋯ n/2) of ν, and the scalar products are meant in L_{2}.

Corollary 15.3. For any even integer p and any N ≥ 1, any sequence X = (X_{j}) of i.i.d. Gaussian variables with values in the space M_{N} of N × N (complex) matrices has its p-th moments defined by pairings. (Here the moments are meant with respect to the functional x → 𝔼 tr(x).)

Proof. Consider the ℝ-linear map φ: M_{N} ⊗ ⋯ ⊗ M_{N} → ℂ defined by

φ(x_{1} ⊗ ⋯ ⊗ x_{p}) = tr(x_{1}x_{2}^{∗} … x_{p−1}x_{p}^{∗}).

Let k = (k_{1}, …, k_{p}) where k_{1}, k_{2}, …, k_{p} are positive integers. Let Y_{k} = (X_{k_{1}}, …, X_{k_{p}}). Applying (15.2) to X_{k_{1}} ⊗ ⋯ ⊗ X_{k_{p}} we find

𝔼φ(X_{k_{1}} ⊗ ⋯ ⊗ X_{k_{p}}) = ∑_{ν∈P_{2}(p)} 𝔼φ(Y_{k}^{⊗ν}).

Since the variables X_{1}, X_{2}, ... are assumed independent, we have 𝔼φ(Y_{k}^{⊗ν}) = 0 except possibly when k ∼ ν (indeed, if {i, j} is a block of ν and k_{i} ≠ k_{j} then the entries of the k_{i} factor of Y_{k}^{⊗ν} are orthogonal to those of the k_{j} factor and independent of all the other factors of Y_{k}^{⊗ν}). Moreover, since X_{1}, X_{2}, ... all have the same distribution, it is easy to check that the distribution of Y_{k}^{⊗ν} depends only on ν and not on k. Thus we obtain

𝔼φ(X_{k_{1}} ⊗ ⋯ ⊗ X_{k_{p}}) = ∑_{ν∼k} ψ(ν)

with ψ(ν) = 𝔼φ(Y^{⊗ν}), with Y = (X_{1}, …, X_{n}).

Remark 15.4. The preceding result is used in [12] for the complex Gaussian random matrices (Y_{j}^{(N)})_{j≥1} appearing in Corollary 11.13. In that case, since Y^{(N)} ⊗ Y^{(N)} has mean zero, there is an extra cancellation: ψ(ν) = 0 for any partition ν ∈ P_{2}(n) admitting a block with two indices of the same parity.

More generally, the same proof shows

Corollary 15.5. The preceding corollary is valid for any even integer p for any i.i.d. Gaussian sequence with values in L_{p}(M, τ) for any non-commutative measure space (M, τ). (Here the moments are meant with respect to x ↦ 𝔼τ(x).)

Using exactly the same method but replacing stochastic independence by freeness in the sense of [35] and Gaussian by semi-circular (or “free-Gaussian”), it is easy to extend the preceding Corollary to the free case. More generally, we can use the q-Fock space (−1 ≤ q ≤ 1) and the associated q-Gaussian variables described in [3, 4, 2]. Fix q with −1 ≤ q < 1. Given a complex Hilbert space H, we denote by F_{q}(H) the q-Fock space associated to H. Let us assume that H is the complexification of a real Hilbert space H so that H = H + iH. For simplicity we assume H = ℓ_{2}(ℝ). To any real Hilbert subspace K ⊂ H we can associate (following [2]) a von Neumann algebra Γ_{q}(K), so that we have natural embeddings Γ_{q}(K_{1}) ⊂ Γ_{q}(K_{2}) when K_{1} ⊂ K_{2}. Moreover Γ_{q}(H) is equipped with a normalized trace (faithful and normal) denoted by τ_{q}, that we may restrict to Γ_{q}(K) to view the latter as a non-commutative probability space. For any h ∈ H we denote by a^{∗}(h) (resp. a(h)) the q-creation (resp. qannihilation) operator on F_{q}(H) and we let g_{q}(h) = a(h)+a^{∗}(h). By definition, the von Neumann algebras Γ_{q}(K) is generated by {g_{q}(h) | h ∈ K}. We will say that a family X_{1}, … X_{n} in Γ_{q}(H) is q-independent if there are mutually orthogonal real subspaces K_{j} ⊂ H such that X_{j} ∈ Γ_{q}(K_{j}). Let B = B(ℓ_{2}). More generally, consider x_{1}, …, x_{n} in B ⊗ Γ_{q}(H). We will say that x_{1}, …, x_{n} are q-independent if there are K_{j} as above such that x_{j} ∈ B ⊗ Γ_{q}(K_{j}) for all j = 1, …, n. The elements of g_{q}(H) = {g_{q}(h) | h ∈ H} will be called q-Gaussian. More generally, an element x ∈ B ⊗ Γ_{q}(H) will be called q-Gaussian if x ∈ B ⊗ g_{q}(H). The q-Gaussian elements satisfy an analogue (called “second quantization”) of the rotational invariance of Gaussian distributions: For any ℝ-isometry T: K → H the families {g_{q}(t) | t ∈ K} and {g_{q}(T t) | t ∈ K} have identical distributions. Here “same distribution” means equality of the moments of all non-commutative monomials. We will denote by T̃: B ⊗ g_{q}(K) → B ⊗ g_{q}(H) the linear map taking b ⊗ g_{q}(t) to b ⊗ g_{q}(T t) (b ∈ B, t ∈ K). In particular, if x ∈ g_{q}(K) and if K_{1} = K_{2} = ⋯ = K_{s} = K we may use the isometry u_{s}: K → K_{1} ⊕ ⋯ ⊕ K_{s} ⊂ H defined by u_{s}(x) = s^{−1/2}(x ⊕ ⋯ ⊕ x) in order to define elements x_{j} in g_{q}(K_{j}) each with the same distribution as x such that x =^{dist} s^{−1/2}(x_{1} + ⋯ + x_{s}). Let x_{1}, …, x_{n} be any sequence in B ⊗ g_{q}(H) and let π be a partition. For any block α of π we give ourselves an isometry u_{α}: H → H_{α} ⊂ H where H_{α} are mutually orthogonal (real) Hilbert subspaces. Then we define a sequence (y_{1}, …, y_{n}) in B ⊗ g_{q}(H) by setting

y_{j} = ũ_{α}x_{j}.  ∀j ∈ α

It is not hard to check that τ̂(y_{1} ˙⊗ ⋯ ˙⊗y_{n}) depends only on x = (x_{1}, …, x_{n}) and π (and not on the u_{α}’s). Therefore we may set

τ̂(x^{π}) =^{def} τ̂(y_{1} ˙⊗ ⋯ ˙⊗y_{n}).

As before, by symmetry τ̂_{q}(x_{1} ˙⊗ ⋯ ˙⊗x_{n}) = 0 for all odd n. The q-Gaussian analogue of (15.1) is as follows:

Proposition 15.6. For any even integer n and any n-tuple x_{1} …, x_{n} in B ⊗ g_{q}(H) (−1 ≤ q < 1), we have

τ̂_{q}(x_{1} ˙⊗ ⋯ ˙⊗x_{n}) = ∑_{ν∈P_{2}(n)} τ̂_{q}(x^{ν}).

Proof. With the preceding ingredients, this can be proved exactly as Proposition 15.1 above.

In particular, replacing B by ℂ, we find

Corollary 15.7. Any q-Gaussian sequence (x_{j}) in the sense of [2, Def. 3.3] with covariance equal to the identity matrix has p-th moment, defined by parings for any even integer p.

Acknowledgment. I am extremely grateful to Quanhua Xu for a careful reading that led to numerous corrections and improvements.

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Gilles Pisier

Texas A&M University College Station TX 77843 U. S. A. and Université Paris VI Equipe d’Analyse Case 186 75252 Paris Cedex 05 France