## Complex interpolation between Hilbert, Banach and operator spaces (2008)

Citations: | 7 - 0 self |

### BibTeX

@MISC{Pisier08complexinterpolation,

author = {Gilles Pisier},

title = {Complex interpolation between Hilbert, Banach and operator spaces },

year = {2008}

}

### OpenURL

### Abstract

Motivated by a question of Vincent Lafforgue, we study the Banach spaces X satisfying the following property: there is a function ε → ∆X(ε) tending to zero with ε> 0 such that every operator T: L2 → L2 with ‖T ‖ ≤ ε that is simultaneously contractive (i.e. of norm ≤ 1) on L1 and on L ∞ must be of norm ≤ ∆X(ε) on L2(X). We show that ∆X(ε) ∈ O(ε α) for some α> 0 iff X is isomorphic to a quotient of a subspace of an ultraproduct of θ-Hilbertian spaces for some θ> 0 (see Corollary 6.7), where θ-Hilbertian is meant in a slightly more general sense than in our previous paper [43]. Let Br(L2(µ)) be the space of all regular operators on L2(µ). We are able to describe the complex interpolation space (Br(L2(µ)),B(L2(µ))) θ. We show that T: L2(µ) → L2(µ) belongs to this space iff T ⊗ idX is bounded on L2(X) for any θ-Hilbertian space X. More generally, we are able to describe the spaces (B(ℓp0),B(ℓp1))θ or (B(Lp0),B(Lp1))θ for any pair 1 ≤ p0,p1 ≤ ∞ and 0 < θ < 1. In the same vein, given a locally compact Abelian group G, let M(G) (resp. PM(G)) be the space of complex measures (resp. pseudo-measures) on

### Citations

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Citation Context ...heir determinants as those of the corresponding matrices. Note that since z ↦→ det(u(z)) is analytic, bounded and non identically zero, we must have det(u(z)) ̸= 0 a.e. on ∂D and (by Jensen, see e.g. =-=[14, 18]-=-) log |det(u(z))| must be in L1(∂D). Moreover, the classical formula for the inverse of a matrix shows (since u is bounded on D) that there is a constant c such that ‖u(z) −1‖ ≤ c|det(u(z))| −1 a.e. o... |

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Citation Context ...M G | ℓ 0 2 (G) ‖ ‖M G f − Ef‖2 ≤ ε(G)‖f‖2. A sequence of k-regular graphs {G(m)} is called expanding if supε(G(m)) < 1 and |G(m)| → ∞ m when m → ∞. In connection with the Baum–Connes conjecture (see =-=[19, 50]-=-) it is of interest to understand in which Banach spaces X we can embed coarsely, but uniformly over m, such a sequence {G(m)} viewed as a sequence of metric spaces. More precisely, we will say that {... |

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Citation Context ... observed that such an embedding is impossible into X if ∆X(ε) → 0 when ε → 0. See §3 for more on this. 4The preceding results all have analogues in the recently developed theory of operator spaces (=-=[15, 47]-=-). Indeed, the author previously introduced and studied mainly in [48, 49] all the necessary ingredients, notably complex interpolation and operator space valued non-commutative Lp-spaces. With these ... |

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Citation Context ...Lp (e.g. those generated by the Laplacian on a Riemannian manifold) or in that of random walks on groups. In this section, we illustrate this by combinatorial Laplacians on expanding graphs (cf. e.g. =-=[36]-=-). 14We refer the reader to the web site http://kam.mff.cuni.cz/∼matousek/metrop.ps for a list of problems (in particular Linial’s contribution) related to this section. See also [39] for a similar t... |

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Citation Context ...r operators Let 1 ≤ p < ∞ throughout this section. For operators on Lp it is well known that the notions of “regular’ and “order bounded” coincide, so we will simply use the term regular. We refer to =-=[50, 72]-=- for general facts on this. The results of this section are all essentially well known, we only recall a few short proofs for the reader’s convenience and to place them in the context that is relevant... |

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Citation Context ... observed that such an embedding is impossible into X if ∆X(ε) → 0 when ε → 0. See §3 for more on this. 4The preceding results all have analogues in the recently developed theory of operator spaces (=-=[15, 47]-=-). Indeed, the author previously introduced and studied mainly in [48, 49] all the necessary ingredients, notably complex interpolation and operator space valued non-commutative Lp-spaces. With these ... |

87 |
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Citation Context ...→ 0. See §3 for more on this. 4The preceding results all have analogues in the recently developed theory of operator spaces ([18, 66]). Indeed, the author previously introduced and studied mainly in =-=[64, 63]-=- all the necessary ingredients, notably complex interpolation and operator space valued non-commutative Lp-spaces. With these tools, it is an easy task to check the generalized statements, so that we ... |

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Citation Context ...Br(Lp(µ),Lp ′(µ′ )). 1.8 We refer e.g. to [59] for more information and references on all this subsection (see also [18, 66] for the operator space analogue). The original ideas can be traced back to =-=[23]-=-. An operator v: E → F between Banach spaces is called nuclear if it can be written as an absolutely convergent series of rank one operators, i.e. there are x ∗ n ∈ E ∗ , yn ∈ F with ∑ ‖x ∗ n‖‖yn‖ < ∞... |

86 | Factorization of Linear Operators and Geometry of Banach Spaces - Pisier - 1986 |

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Citation Context ... us extend ϕ arbitrarily outside C, say by setting ϕ(z) = I for all z outside C. By the compatibility assumption, we have clearly k1(·)I ≤ ϕ(·) ≤ I. By the classical matricial Szegö theorem (cf. e.g. =-=[22]-=-) there is a bounded outer function F : D → Mn such that |F(z)| 2 = F(z) ∗ F(z) = ϕ(z) on ∂D. We now define Y (z) by setting ∀x ∈ C n ‖x‖ Y (z) = ‖F(z) −1 x‖ X(z). Note that for any z in C we have Thi... |

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Citation Context ... for some constant C (independent of n). Thus, for any Banach space X, we have Therefore, we find ‖Γ(n)X‖ ≤ C Log(n + 1)∆X(( Log(n + 1)) −1 ). X uniformly curved ⇒ ‖Γ(n)X‖ ∈ o(Log(n)). As observed in =-=[42]-=-, this implies that X is super-reflexive and hence, by Enflo’s theorem [16], isomorphic to a uniformly convex space. It is natural to wonder about the converse: Problem 2.4. Is is true that uniformly ... |

73 |
The operator Hilbert space OH, complex interpolation and tensor norms
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(Show Context)
Citation Context ...→ 0. See §3 for more on this. 4The preceding results all have analogues in the recently developed theory of operator spaces ([15, 47]). Indeed, the author previously introduced and studied mainly in =-=[48, 49]-=- all the necessary ingredients, notably complex interpolation and operator space valued non-commutative Lp-spaces. With these tools, it is an easy task to check the generalized statements, so that we ... |

71 |
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Citation Context ...aneously on ℓn p0 and ℓnp1 of norm ≤ 1 on ℓn p for any p0 < p < p1, and similarly for operators on Lp-spaces. Later on in 1938, Thorin found the most general form using a complex variable method; see =-=[2]-=- for more on this history. Then around 1960, J.L. Lions and independently A. Calderón [9] invented the complex interpolation method, which may be viewed as a far reaching “abstract” version of the Rie... |

59 |
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Citation Context ...imum runs over all X in B and all factorizations E u2 u1 −→ X −→ F. Let ΓB(E,F) denote the space of those u that admit such a factorization. Then ΓB(E,F) is a vector space and γB is a norm on it. See =-=[32, 41]-=- for details. 4.10 Note that ‖u‖ ≤ γB(u) and equality holds if u has rank 1. Therefore we have for all u: E → F ‖u‖ ≤ γB(u) ≤ N(u). 5 Complex interpolation of families of Banach spaces Let D = {z ∈ C ... |

40 | Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces
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Citation Context ...ng graphs (cf. e.g. [45]). We refer the reader to the web site http://kam.mff.cuni.cz∼matousek/metrop.ps for a list of problems (in particular Linial’s contribution) related to this section. See also =-=[53]-=- for a similar theme. Let G be a finite set equipped with a graph structure, so that G is the vertex set and we give ourselves a symmetric set of edges E ⊂ G × G. We assume the graph G k-regular, i.e.... |

38 |
Lectures on Coarse Geometry
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Citation Context ...M G | ℓ 0 2 (G) ‖ ‖M G f − Ef‖2 ≤ ε(G)‖f‖2. A sequence of k-regular graphs {G(m)} is called expanding if supε(G(m)) < 1 and |G(m)| → ∞ m when m → ∞. In connection with the Baum–Connes conjecture (see =-=[19, 50]-=-) it is of interest to understand in which Banach spaces X we can embed coarsely, but uniformly over m, such a sequence {G(m)} viewed as a sequence of metric spaces. More precisely, we will say that {... |

35 |
Some remarks on Banach spaces in which martingale difference sequences are unconditional,” Arkiv für
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Citation Context ...his). 4.7 If C = {T } where T is the Hilbert transform on Lp(T) (or Lp(R)), then C ◦◦ contains all martingale transforms and a number of Fourier multipliers of Hörmander type. This is due to Bourgain =-=[3]-=-. Conversely if C is the class of all martingale transforms, then it was known before Bourgain’s paper that C ◦◦ contains the Hilbert transform and various singular integrals. This is due to Burkholde... |

34 |
spaces which can be given an equivalent uniformly convex norm.IsraelJ.Math.13(1972
- Enflo, Banach
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Citation Context ...he unit ball and hence (2 −2 ‖x + y‖ 2 + 2 −2 ‖x − y‖ 2 ) 1/2 ≤ ∆X(2 −1/2 )(‖x‖ 2 + ‖y‖ 2 ) 1/2 min{‖2 −1 (x + y)‖, ‖2 −1 (x − y)‖} ≤ ∆X(2 −1/2 ) < 1. Modulo the beautiful results of James and Enflo (=-=[25, 16]-=-) on “super-reflexivity,” this observation shows that any uniformly curved Banach space is isomorphic to a uniformly convex one. Recall that a Banach space is called uniformly convex if δX(ε) > 0 for ... |

31 | A course on Borel sets - Srivastava - 1998 |

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Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients
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Citation Context ...of regular operators. In §4, we describe a certain duality between, on one hand, classes of Banach spaces, and on the other one, classes of operators on Lp. Although these ideas already appeared (cf. =-=[31, 32, 23]-=-), the viewpoint we emphasize was left sort of implicit. We hope to stimulate further research on the list of related questions that we present in this section. In §5, we present background on the com... |

25 |
A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions
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Citation Context ...the class of all martingale transforms, then it was known before Bourgain’s paper that C ◦◦ contains the Hilbert transform and various singular integrals. This is due to Burkholder and McConnell (cf. =-=[11, 10]-=-). This leaves entirely open many interesting questions. For instance, it would be nice (perhaps not so hopeless ?) to have a description of {T } ◦◦ when T is the Hilbert transform on Lp(T), say for p... |

24 |
Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol
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Citation Context ...aves entirely open many interesting questions. For instance, it would be nice (perhaps not so hopeless ?) to have a description of {T } ◦◦ when T is the Hilbert transform on Lp(T), say for p = 2. See =-=[20, 55]-=- for results in this direction. 4.8 In sharp contrast, Problem 4.2 is in some sense already solved by the following theorem of Hernandez [29], extending Kwapién’s ideas in [41]. Kwapień proved that a ... |

18 | Topics in Hardy Classes and Univalent Functions, Birkhäuser - Rosenblum, Rovnyak - 1994 |

17 | Similarity problems and completely bounded maps. Second, expanded edition - Pisier - 2001 |

16 |
G.: Bounded linear operators between C ∗ -algebras
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Citation Context ...rator space structure. The space K0 ∩ K1 ∩ · · · ∩ Kd has already appeared in Harmonic Analysis over F∞, the free group with countably infinitely many generators {gi}. Indeed, it was shown in [4] (in =-=[20]-=- for the d = 1 case) that K0 ∩ · · · ∩Kd can be identified with the closed span in C∗ λ (F∞) of {λ(gi1gi2 ...gid ) | i1,i2,... ,id ∈ N}. For an extension of this to the non-commutative Lp-space over F... |

16 |
The theory of p-spaces with an application to convolution operators
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Citation Context ... and classes of operators on Lp on the other hand. The general ideas underlying this are already implicitly in Kwapién’s remarkable papers [31, 32] and Roberto Hernandez’s thesis (cf. [23]). See also =-=[24, 5, 6, 53]-=-. We seize the occasion to develop this theme more explicitly in this paper. Fix 1 ≤ p ≤ ∞. Consider a class of operators C acting between Lp-spaces. A typical element of C is a (bounded) operator T :... |

15 |
On embedding expanders into ℓp spaces
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Citation Context ...the appendix of [52] for the case of Banach lattices of finite cotype, or spaces with a unit ball uniformly homeomorphic to a subset of ℓ2. We refer to [49] for a discussion of type and cotype and to =-=[51, 48]-=- for a survey of recent work on coarse embeddability between Banach spaces. 4 A duality operators/classes of Banach spaces In this section, we describe a “duality” (or a “polarity”) between classes of... |

14 |
Un renforcement de la propriété
- LAFFORGUE
(Show Context)
Citation Context ...ntain uniformly coarsely an expanding sequence. Proof. Indeed, if ε = supε(G(m)) < 1 and X is uniformly curved, then 2∆X(εn /2) → 0 when m n → ∞ so we can always choose n so that δ < 1. ) 1/2 . 16In =-=[34]-=-, in answer to a question of Naor, V. Lafforgue shows that certain specific expanding sequences do not embed uniformly coarsely even in any Banach space with non trivial type. This is a stronger state... |

13 |
Type, cotype and K-convexity. Handbook of the geometry of Banach spaces
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(Show Context)
Citation Context ...viewed here, exceptionally, as a real Banach space). We say X ⊃ E almost isometrically if for any ε > 0 there is a subspace ˜ E ⊂ X with Banach–Mazur distance d(E, ˜ E) < 1 + ε. By known results (see =-=[38]-=-), this shows that X curved ⇒ X of type p for some p > 1. However, a stronger conclusion can be reached using the Hilbert transform, say in its simplest discrete matricial form. Fix n ≥ 1. Let Γ(n) = ... |

11 | On the interpolation of injective or projective tensor products of Banach spaces
- Kouba
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(Show Context)
Citation Context ...onal operator spaces E,F. Let cγ(z) = Γ cB(z)(E,F). Then for all w: E → F we have ∀ξ ∈ D ‖w‖ ΓcB(ξ)(E,F) ≤ ‖w‖ cγ(ξ). 14 Examples with the Haagerup tensor product Using Kouba’s interpolation theorem (=-=[30]-=-) we will produce some interesting examples of compatible families of operator spaces involving quite naturally more than 2 spaces. Let J(1),J(2),...,J(d) be arbitrary measurable subsets of ∂D. We wil... |

10 |
Some self-dual properties of normed linear spaces, in Symp
- James
- 1972
(Show Context)
Citation Context ...he unit ball and hence (2 −2 ‖x + y‖ 2 + 2 −2 ‖x − y‖ 2 ) 1/2 ≤ ∆X(2 −1/2 )(‖x‖ 2 + ‖y‖ 2 ) 1/2 min{‖2 −1 (x + y)‖, ‖2 −1 (x − y)‖} ≤ ∆X(2 −1/2 ) < 1. Modulo the beautiful results of James and Enflo (=-=[25, 16]-=-) on “super-reflexivity,” this observation shows that any uniformly curved Banach space is isomorphic to a uniformly convex one. Recall that a Banach space is called uniformly convex if δX(ε) > 0 for ... |

10 |
Some applications of the complex interpolation method to Banach lattices
- Pisier
- 1979
(Show Context)
Citation Context ...rphic to a quotient of a subspace of an ultraproduct of θ-Hilbertian spaces for some θ > 0 (see Corollary 6.7), where θ-Hilbertian is meant in a slightly more general sense than in our previous paper =-=[43]-=-. Let Br(L2(µ)) be the space of all regular operators on L2(µ). We are able to describe the complex interpolation space (Br(L2(µ)),B(L2(µ))) θ . We show that T : L2(µ) → L2(µ) belongs to this space if... |

10 |
Complex interpolation and regular operators between Banach lattices
- Pisier
- 1994
(Show Context)
Citation Context ...nal (or merely reflexive). In that case Qp(X) = Tp ′(X∗ ) ∗ , so that ϕX extends to a bounded bilinear form on Sp(X) × Tp ′(X∗ ) iff uX : Sp(X) → Qp(X) is bounded. The following extends a result from =-=[44]-=-. Theorem 10.1. Let u: Sp → Qp be as above. Fix C ≥ 1. The following are equivalent. (i) There is a regular operator ũ: Lp(µ) → Lp(µ ′ ) with ‖ũ‖reg ≤ C such that u = qũ |Sp . (ii) For any Banach spac... |

10 | Non-commutative Khintchine type inequalities associated with free groups
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(Show Context)
Citation Context ...ive case. Lastly, in §14, we describe a family of operator spaces closely connected to various works on the “non-commutative Khintchine inequalities” for homogeneous polynomials of degree d (see e.g. =-=[54]-=-). Here we specifically need to consider a family {X(z) | z ∈ ∂D} taking (d + 1)-values but we are able to compute precisely the interpolation at the center of D (or at any point inside D). 1 Prelimin... |

8 |
A maximal function characterization of the class H p
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- 1971
(Show Context)
Citation Context ...the class of all martingale transforms, then it was known before Bourgain’s paper that C ◦◦ contains the Hilbert transform and various singular integrals. This is due to Burkholder and McConnell (cf. =-=[8, 7]-=-). This leaves entirely open many interesting questions. For instance, it would be nice (perhaps not so hopeless ?) to have a description of {T } ◦◦ when T is the Hilbert transform on Lp(T), say for p... |

8 |
Complex interpolation for families of Banach spaces. Harmonic Analysis
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(Show Context)
Citation Context ...undary of a complex domain D and not only pairs of Banach spaces This generalization was developed around 1980 in a series of papers mainly by Coifman, Cwikel, Rochberg, Sagher, Semmes and Weiss (cf. =-=[10, 11, 12, 51, 13]-=-). There ∂D can be the unit circle and, restricting to the n-dimensional case for simplicity, we may take Bz = (Cn, ‖ ‖z) where {‖ ‖z | z ∈ ∂D} is a measurable family of norms on Cn (with a suitable n... |

8 |
Fourier analysis, Schur multipliers on S p and non-commutative Λ(p)-sets
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- 1999
(Show Context)
Citation Context ...oduct argument and pointwise compactness of the unit balls of the spaces of Schur multipliers under consideration. We skip the details. Remark 7.5. In answer to a question of Peller, it was proved in =-=[21]-=- (see also [55]) that, for any 1 < p < ∞, the interpolation space considered in (7.4) does not contain the space of Schur multipliers that are bounded on Sp. The corresponding fact for Fourier multipl... |

7 |
Interpolation of Banach spaces, Perron processes and
- Coifman, Semmes
(Show Context)
Citation Context ...undary of a complex domain D and not only pairs of Banach spaces This generalization was developed around 1980 in a series of papers mainly by Coifman, Cwikel, Rochberg, Sagher, Semmes and Weiss (cf. =-=[10, 11, 12, 51, 13]-=-). There ∂D can be the unit circle and, restricting to the n-dimensional case for simplicity, we may take Bz = (Cn, ‖ ‖z) where {‖ ‖z | z ∈ ∂D} is a measurable family of norms on Cn (with a suitable n... |

7 |
Vector-valued Hausdorff-Young inequality and applications, Geometric Aspects
- Bourgain
(Show Context)
Citation Context ...f the so-called K-convex spaces, that coincides with the class of spaces that are of type p for some p > 1, or equivalently do not contain ℓn 1 ’s uniformly (see [49] for all this). Bourgain’s papers =-=[4, 5, 6]-=- on the Hausdorff-Young inequality in the Banach space valued case are also somewhat related to the same theme. 4.7 If C = {T } where T is the Hilbert transform on Lp(T) (or Lp(R)), then C ◦◦ contains... |

7 |
Singular integral operators: a martingale approach
- Figiel
- 1989
(Show Context)
Citation Context ...aves entirely open many interesting questions. For instance, it would be nice (perhaps not so hopeless ?) to have a description of {T } ◦◦ when T is the Hilbert transform on Lp(T), say for p = 2. See =-=[20, 55]-=- for results in this direction. 4.8 In sharp contrast, Problem 4.2 is in some sense already solved by the following theorem of Hernandez [29], extending Kwapién’s ideas in [41]. Kwapień proved that a ... |

7 |
Factorization theory for spaces of operators
- Junge
- 1996
(Show Context)
Citation Context ...of regular operators. In §4, we describe a certain duality between, on one hand, classes of Banach spaces, and on the other one, classes of operators on Lp. Although these ideas already appeared (cf. =-=[40, 41, 29, 35]-=-), the viewpoint we emphasize was left sort of implicit. We hope to stimulate further research on the list of related questions that we present in this section. In §5, we present background on the com... |

6 |
On the relation between the two complex methods of interpolation
- Bergh
- 1979
(Show Context)
Citation Context ...ch space Br(L2(µ)) of all regular operators T on L2(µ) (i.e. those T with a kernel (T(s,t)) such that |T(s,t)| is bounded on L2(µ)), then we are able to describe the space (Br(L2(µ)),B(L2(µ))) θ . By =-=[1]-=- this also yields (B0,B1)θ as the closure of B0 ∩ B1 in (B0,B1) θ . The origin of this paper is a question raised by Vincent Lafforgue: what are the Banach spaces X satisfying the following property: ... |

6 |
Norm of convolution by operator-valued functions on free groups
- Buchholz
(Show Context)
Citation Context ...umn) operator space structure. The space K0 ∩ K1 ∩ · · · ∩ Kd has already appeared in Harmonic Analysis over F∞, the free group with countably infinitely many generators {gi}. Indeed, it was shown in =-=[4]-=- (in [20] for the d = 1 case) that K0 ∩ · · · ∩Kd can be identified with the closed span in C∗ λ (F∞) of {λ(gi1gi2 ...gid ) | i1,i2,... ,id ∈ N}. For an extension of this to the non-commutative Lp-spa... |

6 |
Completely bounded maps between sets of Banach spaces operators
- Pisier
(Show Context)
Citation Context ...this are 17already implicitly in Kwapién’s remarkable papers [40, 41] and Roberto Hernandez’s thesis (cf. [29]). See also [31, 8, 9, 77]. In connection with the notion of p-complete boundedness from =-=[60]-=-, we refer the interested reader to [35, Th 1.2.3.7 and Cor. 1.2.4.7] for an even more general duality. We seize the occasion to develop this theme more explicitly in this paper. Fix 1 ≤ p ≤ ∞. Consid... |

5 |
Regular operators between non-commutative Lp-spaces
- Pisier
- 1995
(Show Context)
Citation Context ...by Mn(B(H)). We refer to [15, 47] for general background on operator spaces, completely bounded (in short c.b.) maps, to [48] for notions on non-commutative vector-valued Lp-spaces used below, and to =-=[45]-=- for the non-commutative version of “regular operators” on the Hilbert–Schmidt class S2 or on more general non-commutative Lp. We first give a presentation parallel to §6. Let Sp denote the Schatten p... |

5 |
Interpolation of Banach spaces, differential geometry, and differential equations
- Semmes
- 1988
(Show Context)
Citation Context ...undary of a complex domain D and not only pairs of Banach spaces This generalization was developed around 1980 in a series of papers mainly by Coifman, Cwikel, Rochberg, Sagher, Semmes and Weiss (cf. =-=[10, 11, 12, 51, 13]-=-). There ∂D can be the unit circle and, restricting to the n-dimensional case for simplicity, we may take Bz = (Cn, ‖ ‖z) where {‖ ‖z | z ∈ ∂D} is a measurable family of norms on Cn (with a suitable n... |

4 |
On operators factorizable through Lp space, Actes du Colloque d’Analyse Fonctionnelle de Bordeaux (Univ. de
- Kwapień
- 1971
(Show Context)
Citation Context ...of regular operators. In §4, we describe a certain duality between, on one hand, classes of Banach spaces, and on the other one, classes of operators on Lp. Although these ideas already appeared (cf. =-=[31, 32, 23]-=-), the viewpoint we emphasize was left sort of implicit. We hope to stimulate further research on the list of related questions that we present in this section. In §5, we present background on the com... |

4 |
Integral Operators and Changes of Density
- Weis
- 1982
(Show Context)
Citation Context ...density argument to replace fully contractive operators by regular ones in the definition of ∆X. This kind of argument (related to the classical “Schur test” and its converse) is well known, see e.g. =-=[29, 27, 54]-=-. Proposition 2.1. Let X be a Banach space. Fix 0 < ε < 1 and δ > 0. The following are equivalent: (i) For any n and any n × n matrix T = [aij] with ‖T : ℓn 2 → ℓn2 ‖ ≤ ε and such that ‖T : ℓn2 → ℓn 2... |

4 |
A Hausdorff-Young inequality for B-convex Banach spaces
- BOURGAIN
- 1982
(Show Context)
Citation Context ...f the so-called K-convex spaces, that coincides with the class of spaces that are of type p for some p > 1, or equivalently do not contain ℓn 1 ’s uniformly (see [49] for all this). Bourgain’s papers =-=[4, 5, 6]-=- on the Hausdorff-Young inequality in the Banach space valued case are also somewhat related to the same theme. 4.7 If C = {T } where T is the Hilbert transform on Lp(T) (or Lp(R)), then C ◦◦ contains... |