Non-commutative vector valued $L_p$-spaces and completely $p$-summing map

Non-commutative vector valued

L_{p}

-spaces and completely

p

-summing map

Pisier, Gilles

Astérisque, no. 247 (1998) , 137 p.

MR Zbl | 6 citations dans Numdam

@book{AST_1998__247__R1_0,
     author = {Pisier, Gilles},
     title = {Non-commutative vector valued $L_p$-spaces and completely $p$-summing map},
     series = {Ast\'erisque},
     year = {1998},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {247},
     mrnumber = {1648908},
     zbl = {0937.46056},
     language = {en},
     url = {https://www.numdam.org/item/AST_1998__247__R1_0/}
}

TY  - BOOK
AU  - Pisier, Gilles
TI  - Non-commutative vector valued $L_p$-spaces and completely $p$-summing map
T3  - Astérisque
PY  - 1998
IS  - 247
PB  - Société mathématique de France
UR  - https://www.numdam.org/item/AST_1998__247__R1_0/
LA  - en
ID  - AST_1998__247__R1_0
ER  -

%0 Book
%A Pisier, Gilles
%T Non-commutative vector valued $L_p$-spaces and completely $p$-summing map
%S Astérisque
%D 1998
%N 247
%I Société mathématique de France
%U https://www.numdam.org/item/AST_1998__247__R1_0/
%G en
%F AST_1998__247__R1_0

Pisier, Gilles. Non-commutative vector valued $L_p$-spaces and completely $p$-summing map. Astérisque, no. 247 (1998), 137 p. https://www.numdam.org/item/AST_1998__247__R1_0/

Bibliographie
Cité par

[BL] J. Bergh and J. Löfström. Interpolation spaces. An introduction. Springer Verlag, New York. 1976. | MR | Zbl | DOI

[B1] D. Blecher. Tensor products of operator spaces II. Canadian J. Math. 44 (1992) 75-90. | MR | Zbl | DOI

[B2] D. Blecher. The standard dual of an operator space. Pacific J. Math. 153 (1992) 15-30. | MR | Zbl | DOI

[B3] D. Blecher. Generalizing Grothendieck's program, in "Function spaces", Edited by K. Jarosz, Lecture Notes in Pure and Applied Math. vol. 136, Marcel Dekker, 1992. | MR | Zbl

[BP] D. Blecher and V. Paulsen. Tensor products of operator spaces. J. Funct. Anal. 99 (1991) 262-292. | MR | Zbl | DOI

[BS] D. Blecher and R. Smith. The dual of the Haagerup tensor product. Journal London Math. Soc. 45 (1992) 126-144. | MR | Zbl | DOI

[Bo1] J. Bourgain, Vector valued singular integrals and the $H^{1} - B M O$ duality, Probability theory and harmonic analysis (Chao-Woyczynski, ed.), pp. 1-19; Dekker, New York, 1986. | Zbl

[Bo2] J. Bourgain. Some remarks on Banach spaces in which martingale differences are unconditional, Arkiv för Math. 21 (1983) 163-168. | MR | Zbl | DOI

[Bo3] J. Bourgain. Bounded orthogonal systems and the $Λ (p)$ -set problem. Acta Math. 162 (1989) 227-245. | MR | Zbl | DOI

[Bon1] A. Bonami. Ensembles $Λ (p)$ dans le dual de $D^{\infty}$ . Ann. Inst. Fourier, Grenoble, 18 (1968) 193-204. | MR | Zbl | EuDML | Numdam | DOI

[Bon2] A. Bonami. Étude des coefficients de Fourier des fonctions de $L^{p} (G)$ . Ann. Inst. Fourier, Grenoble, 20 (1970) 335-402. | MR | Zbl | EuDML | Numdam | DOI

[Boz1] M. Bożejko. The existence of $Λ (p)$ sets in discrete noncommutative groups. Bolletino U.M.I. 11 (1973) 198-199. | MR | Zbl

[Boz2] M. Bożejko. A remark to my paper (The existence of $Λ (p)$ sets in discrete noncommutative groups). Bolletino U.M.I. 8 (1975) 579-582. | MR | Zbl

[Boz3] M. Bożejko. On $Λ (p)$ sets with minimal constant in discrete noncommutative groups. Proc. Amer. Math. Soc. 51 (1975) 407-412. | MR | Zbl | DOI

[BR] O. Bratelli and D. Robinson. Operator algebras and quantum statistical mechanics II. Springer Verlag, New-York, 1981. | MR | Zbl

[Bu1] D. Burkholder. Distribution function inequalities for martingales. Ann. Prob. 1 (1973) 19-42. | MR | Zbl | DOI

[Bu2] D. Burkholder. A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9 (1981) 997-1011. | MR | Zbl | DOI

[BV] H. Bercovici and D. Voiculescu. Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42 (1993) 733-773. | MR | Zbl | DOI

[Ca] A. Calderón. Intermediate spaces and interpolation, the complex method. Studia Math. 24 (1964) 113-190. | MR | Zbl | EuDML | DOI

[Co] A. Connes. Classification of injective factors, Cases $I I_{1}$ , $I I_{\infty}$ $I I I_{λ}$ $λ \neq 1$ . Ann. Math. 104 (1976) 73-116. | MR | Zbl

[DF] A. Defant, K. Floret. Tensor norms and operator ideals. North-Holland, Amsterdam, 1993. | MR | Zbl

[DU] J. Diestel and J. Uhl. Vector measures. Amer. Math. Soc. Math. Surveys 15, Providence, 1977. | MR | Zbl

[Di] J. Dixmier. Formes linéaires sur un anneau d'opérateurs. Bull. Soc. Math. France 81 (1953) 9-39. | MR | Zbl | EuDML | Numdam | DOI

[DCH] J. De Cannière and U. Haagerup. Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups. Amer. J. Math. 107 (1985) 455-500. | MR | Zbl | DOI

[DJT] J. Diestel, H. Jarchow and A. Tonge. Absolutely summing operators. Cambridge University press, 1995. | MR | Zbl | DOI

[E] E. Effros. Advances in quantized functional analysis. Proceedings International Congress of Mathematicians, Berkeley 1986, p. 906-916. | MR | Zbl

[EE] E. Effros and R. Exel. On multilinear double commutant theorems. Operator algebras and applications, Vol. 1, London Math. Soc. Lecture Notes Series 135 (1988) 81-94. | MR | Zbl

[EH] E. Effros and U. Haagerup. Lifting problems and local reflexivity for $C^{*}$ -algebras. Duke Math. J. 52 (1985) 103-128. | MR | Zbl | DOI

[EK] E. Effros and A. Kishimoto. Module maps and Hochschild-Johnson cohomology. Indiana Univ. Math. J. 36 (1987) 257-276. | MR | Zbl | DOI

[EKR] E. Effros, J. Kraus and Z. J. Ruan. On two quantized tensor products. Preprint 1992. To appear. | Zbl | MR

[ER1] E. Effros and Z. J. Ruan. On matricially normed spaces. Pacific J. Math. 132 (1988) 243-264. | MR | Zbl | DOI

[ER2] E. Effros and Z. J. Ruan. A new approach to operator spaces. Canadian Math. Bull. 34 (1991) 329-337. | MR | Zbl | DOI

[ER3] E. Effros and Z. J. Ruan. On the abstract characterization of operator spaces. Proc. Amer. Math. Soc. 119 (1993) 579-584. | MR | Zbl | DOI

[ER4] E. Effros and Z. J. Ruan. Self duality for the Haagerup tensor product and Hilbert space factorization. J. Funct. Anal. 100 (1991) 257-284. | MR | Zbl | DOI

[ER5] E. Effros and Z. J. Ruan. Recent development in operator spaces. Current Topics in Operator Algebras. Proceedings of the ICM-90 Satellite Conference held in Nara (August 1990). World Sci. Publishing, River Edge, N.J., 1991, p. 146-164. | MR | Zbl

[ER6] E. Effros and Z. J. Ruan. Mapping spaces and liftings for operator spaces. Proc. London Math. Soc. 69 (1994) 171-197. | MR | Zbl | DOI

[ER7] E. Effros and Z. J. Ruan. The Grothendieck-Pietsch and Dvoretzky-Rogers Theorems for operator spaces. J. Funct. Anal. 122 (1994) 428-450. | MR | Zbl | DOI

[ER8] E. Effros and Z. J. Ruan. On approximation properties for operator spaces. International J. Math. 1 (1990) 163-187. | MR | Zbl | DOI

[FaK] T. Fack and H. Kosaki. Generalized $s$ -numbers of $τ$ -measurable operators. Pacific J. Math. 123 (1986) 269-300. | MR | Zbl | DOI

[Fi1] F. Fidaleo. Operator space structures and the split property. J. Operator Theory 31 (1994) 207-218. | MR | Zbl

[Fi2] F. Fidaleo. Some operator ideals in non-commutative functional analysis. Preprint to appear. | Zbl | MR | DOI

[G] A. Grothendieck. Résumé de la théorie métrique des produits tensoriels topologiques. Boll. Soc. Mat. Sao-Paulo 8 (1956) 1-79. | Zbl | MR

[Gr] U. Groh. Uniform ergodic theorems for identity preserving Schwarz maps on W-algebras. J. Operator Theory 11 (1984) 395-404. | MR | Zbl

[H1] U. Haagerup. Injectivity and decomposition of completely bounded maps in "Operator algebras and their connection with Topology and Ergodic Theory". Springer Lecture Notes in Math. 1132 (1985) 170-222. | MR | Zbl | DOI

[H2] U. Haagerup. $L^{p}$ -spaces associated with an arbitrary von Neumann algebra. Algèbres d'opérateurs et leurs applications en physique mathématique. (Colloque CNRS, Marseille, juin 1977) Editions du CNRS, Paris 1979. | Zbl

[H3] U. Haagerup. Decomposition of completely bounded maps on operator algebras. Unpublished manuscript. Sept. 1980.

[H4] U. Haagerup. An example of a non-nuclear $C^{*}$ -algebra which has the metric approximation property. Inventiones Math. 50 (1979) 279-293. | Zbl | EuDML | DOI

[HP1] U. Haagerup and G. Pisier. Factorization of analytic functions with values in non-commutative $L_{1}$ -spaces. Canadian Journal of Math. 41 (1989) 882-906. | Zbl | DOI

[HP2] U. Haagerup and G. Pisier. Bounded linear operators between $C^{*}$ -algebras. Duke Math. J. 71 (1993) 889-925. | Zbl | DOI

[Ha] A. Harcharras. Fourier analysis, Schur multipliers on $S_{p}$ and non-commutative $Λ (p)$ -sets. Thesis. Université Paris 6. 1997. | Zbl

[Hei] S. Heinrich. Ultraproducts in Banach space theory. J. für die reine und Angew. Math. 313 (1980) 72-104. | Zbl | EuDML

[Her] C. Herz. The theory of $p$ -spaces with an application to convolution operators. Trans. Amer. math. Soc. 154 (1971) 69-82. | Zbl

[Hi] M. Hilsum. Les espaces $L^{p}$ d'une algèbre de von Neumann définis par la dérivée spatiale. J. Funct. Anal. 40 (1981) 151-169. | Zbl | DOI

[HR] H. Halberstam and K. Roth. Sequences, volume I. Oxford University Press, 1966. | Zbl

[Ju] M. Junge. Factorization theory for spaces of operators. Habilitation thesis. Kiel university, 1996.

[KR] R. Kadison and J. Ringrose. Fundamentals of the theory of operator algebras, Vol. II, Advanced Theory, Academic Press, New-York 1986. | Zbl

[Ki] E. Kirchberg. On subalgebras of the CAR-algebra. J. Funct. Anal. 129 (1995) 35-63. | Zbl | DOI

[Ko] H. Kosaki. Applications of the complex interpolation method to a von Neumann algebra: non commutative $L^{p}$ -spaces. J. Funct. Anal. 56 (1984) 29-78. | Zbl | DOI

[Kr] J. Kraus. The slice map problem and approximation properties. J. Funct. Anal. 102 (1991) 116-155. | Zbl | DOI

[Ku] R. Kunze. $L_{p}$ Fourier transforms on locally compact unimodular groups. Trans. Amer. Math. Soc. 89 (1958) 519-540. | Zbl

[Kw1] S. Kwapień. A linear topological characterization of inner product spaces. Studia Math. 38 (1970) 277-278. | Zbl | EuDML | DOI

[Kw2] S. Kwapień. On operators factorizable through $L_{p}$ -spaces. Bull. Soc. Math. France, Mémoire 31-32 (1972) 215-225. | Zbl | EuDML | Numdam

[Le] M. Leinert. Faltungsoperatoren auf gewissen diskreten gruppen. Studia Math. 52 (1974) 149-158. | Zbl | EuDML | DOI

[L] E. Lieb. Convex trace functions and the Wigner-Yanase-Dyson conjecture. Form convex functions and the $W Y D L$ and other inequalities. Advances in Math. 11 (1973) 267-288. | Zbl | DOI

[LP] J. Lindenstrauss and A. Pełczyński. Absolutely summing operators between $L_{p}$ -spaces and their applications. | Zbl

[LT1] J. Lindenstrauss and L. Tzafriri. Classical Banach spaces. Part I. Springer Verlag, Berlin Heidelberg 1977. | Zbl

[LoR] J. Lopez and K. Ross. Sidon sets. Marcel Dekker. New-York, 1975. | Zbl

[Lu] F. Lust-Piquard. Inégalités de Khintchine dans $C_{p} (1 < p < \infty)$ . C.R. Acad. Sci. Paris 303 (1986), 289-292. | Zbl

[LuP] F. Lust-Piquard and G. Pisier. Non-commutative Khintchine and Paley inequalities. Ark. för Mat. 29 (1991) 241-260. | Zbl | DOI

[Ma] L. Maligranda. Interpolation between sum and intersection of Banach spaces. J. Approx. Theory 47 (1986) 42-53. | MR | Zbl | DOI

[MaP] B. Maurey and G. Pisier. Séries de variables aléatoires vectorielles indépendantes and propriétés géométriques des espaces de Banach. Studia Math. 58 (1976) 45-90. | MR | Zbl | EuDML | DOI

[N] E. Nelson. Notes on non-commutative integration. J. Funct. Anal. 15 (1974) 103-116. | MR | Zbl | DOI

[Pa1] V. Paulsen. Completely bounded maps and dilations. Pitman Research Notes 146. Pitman Longman (Wiley) 1986. | MR | Zbl

[Pa2] V. Paulsen. Representation of Function algebras, Abstract operator spaces and Banach space Geometry. J. Funct. Anal. 109 (1992) 113-129. | MR | Zbl | DOI

[PS] V. Paulsen and R. Smith. Multilinear maps and tensor norms on operator systems. J. Funct. Anal. 73 (1987) 258-276. | MR | Zbl | DOI

[Ped] G. Pedersen. $C^{*}$ -algebras and their automorphisms groups. Academic Press, London New-York, 1979. | MR

[Pe1] V. Peller. Hankel operators of class $S_{p}$ and their applications. Math. Sbornik 41 (1982) 443-479. | Zbl

[Pe2] V. Peller. Vectorial Hankel operators, commutators and related operators of the Schatten-von Neumann class $S_{p}$ . Integral Equations and Operator Theory 5 (1982) 244-272. | MR | Zbl | DOI

[Pi] A. Pietsch. Absolut $p$ -summierende Abbildungen in normierten Räumen. Studia Math. 28 (1967) 333-353. | MR | Zbl | EuDML | DOI

[P1] G. Pisier. The operator Hilbert space $O H$ , complex interpolation and tensor norms. Memoirs Amer. Math. Soc. vol. 122, 585 (1996) 1-103. | MR | Zbl | DOI

[P2] G. Pisier. Factorization of linear operators and the Geometry of Banach spaces. CBMS (Regional conferences of the A.M.S.) 60, (1986), Reprinted with corrections 1987. | MR | Zbl | DOI

[P3] G. Pisier. Complex interpolation and regular operators between Banach lattices. Archiv der Math. (Basel) 62 (1994) 261-269. | MR | Zbl | DOI

[P4] G. Pisier. Dvoretzky's theorem for operator spaces and applications. Houston J. Math. 22 (1996) 399-416. | Zbl | MR

[P5] G. Pisier. Espaces $L_{p}$ non commutatifs à valeurs vectorielles et applications p-sommantes. C. R. Acad. Sci. Paris, 316 (1993) 1055-1060. | MR | Zbl

[P6] G. Pisier. Exact operator spaces. Colloque sur les algèbres d'opérateurs, in "Recent advances in operator algebras" (Orléans 1992) Astérisque (Soc. Math. France) 232 (1995) 159-186. | MR | Zbl | Numdam

[P7] G. Pisier. Martingales with values in uniformly convex spaces. Israel J. Math. 20 (1975) 326-350. | MR | Zbl | DOI

[P8] G. Pisier. Multipliers and lacunary sets in non amenable groups. Amer. J. Math. 117 (1995) 337-376. | MR | Zbl | DOI

[P9] G. Pisier. Similarity problems and completely bounded maps. Springer Lecture Notes 1618 (1995). | Zbl | MR

[P10] G. Pisier. Regular operators between non-commutative $L_{p}$ -spaces. Bull. Sci. Math. 119 (1995) 95-118. | MR | Zbl

[P11] G. Pisier. Holomorphic semi-groups and the geometry of Banach spaces. Annals of Math. 115 (1982) 375-392. | MR | Zbl | DOI

[P12] G. Pisier. Some results on Banach spaces without local unconditional structure. Compositio Math. 37 (1978) 3-19. | MR | Zbl | EuDML | Numdam

[P13] G. Pisier. Riesz transforms: a simpler analytic proof of P. A. Meyer's inequality. Séminaire de Probabilités XXII. Springer Lecture Notes n° 1321 (1988). | MR | Zbl | EuDML | Numdam | DOI

[PX1] G. Pisier and Q. Xu. Inégalités de martingales non commutatives. C. R. Acad. Sci. Paris 323 (1996) 817-822. | MR | Zbl

[PX2] G. Pisier and Q. Xu. Non-commutative martingale inequalities. Comm. Math. Physics 189 (1997) 667-698. | MR | Zbl | DOI

[PW] W. Pusz and S. Woronowicz. Form convex functions and the $W Y D L$ and other inequalities. Letters in Math. Phys. 2 (1978) 505-512. | MR | Zbl | DOI

[RW] A. R. Robertson and S. Wasserman. Completely bounded isomorphisms of injective operator systems. Bull. London Math. Soc. 21 (1989) 285-290. | MR | Zbl | DOI

[Ru] Z. J. Ruan. Subspaces of $C^{*}$ -algebras. J. Funct. Anal. 76 (1988) 217-230. | MR | Zbl | DOI

[Rn] M. Rudelson. Random vectors in isotropic position. Preprint, to appear. | Zbl | DOI

[Rud1] W. Rudin. Fourier analysis on groups. Interscience, New-York, 1962. | MR | Zbl

[Rud2] W. Rudin. Trigonometric series with gaps. J. of Math. and Mechanics 9 (1960) 203-228. | MR | Zbl

[S] I. Segal. A non-commutative extension of abstract integration. Ann. of Math. 57 (1953) 401-457. | MR | Zbl | DOI

[Sk] G. Skandalis. Algèbres de von Neumann de groupes libres and probabilités non commutatives. Séminaire Bourbaki 92/93. Exposé 764. Astérisque, Soc. Math. France. | MR | Zbl | EuDML | Numdam

[St] W. Stinespring. Integration theorems for gages and duality for unimodular groups. Trans. Amer. Math. Soc. 90 (1959) 15-26. | MR | Zbl | DOI

[Ta] M. Takesaki. Theory of Operator Algebras I. Springer-Verlag New-York 1979. | MR | Zbl | DOI

[Te1] M. Terp. Interpolation spaces between a von Neumann algebra and its predual. J. Operator Th. 8 (1982) 327-360. | MR | Zbl

[Te2] M. Terp. $L^{p}$ -spaces associated with von Neumann algebras. Preprint. Copenhagen University. June 1981.

[TJ1] N. Tomczak-Jaegermann. Banach-Mazur distances and finite dimensional operator ideals. Pitman, 1988. | MR | Zbl

[TJ2] N. Tomczak-Jaegermann. The moduli of convexity and smoothness and the Rademacher averages of trace class $S_{p}$ . Studia Math. 50 (1974) 163-182. | MR | Zbl | EuDML

[VDN] D. Voiculescu, K. Dykema, A. Nica. Free random variables. CRM Monograph Series, Vol. 1, Amer. Math. Soc, Providence RI. | MR | Zbl

[W] S. Wasserman. On tensor products of certain group $C^{*}$ -algebras. J. Funct. Anal. 23 (1976) 239-254. | MR | Zbl | DOI

[Wa] F. Wattbled. Interpolation complexe d'un espace de Banach et de son antidual. Comptes Rendus Acad. Sci. Paris 321 (1995) 1437-1440. | MR | Zbl

[X] Q. Xu. Interpolation of operator spaces. J. Funct. Anal. 139 (1996) 500-539. | MR | Zbl | DOI

[Z] A. Zygmund. On Fourier coefficients and transforms of functions of two variables. Studia Math. 50 (1974) 189-201. | MR | Zbl | EuDML | DOI