Duke Mathematical Journal

Rosenthal's theorem for subspaces of noncommutative $L_p$

Marius Junge and Javier Parcet

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Abstract

We show that a reflexive subspace of the predual of a von Neumann algebra embeds into a noncommutative $L_p$-space for some $p>1$. This is a noncommutative version of Rosenthal's result for commutative $L_p$-spaces. Similarly for $1 \le q \lt 2$, an infinite-dimensional subspace $X$ of a noncommutative $L_q$-space either contains $\ell_q$ or embeds in $L_p$ for some $q \lt p \lt 2$. The novelty in the noncommutative setting is a double-sided change of density

Article information

Source
Duke Math. J. Volume 141, Number 1 (2008), 75-122.

Dates
First available in Project Euclid: 4 December 2007

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1196794291

Digital Object Identifier
doi:10.1215/S0012-7094-08-14112-2

Mathematical Reviews number (MathSciNet)
MR2372148

Zentralblatt MATH identifier
1176.46059

Subjects
Primary: 46L53: Noncommutative probability and statistics
Secondary: 46B25: Classical Banach spaces in the general theory

Citation

Junge, Marius; Parcet, Javier. Rosenthal's theorem for subspaces of noncommutative L p . Duke Math. J. 141 (2008), no. 1, 75--122. doi:10.1215/S0012-7094-08-14112-2. http://projecteuclid.org/euclid.dmj/1196794291.


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References

  • D. J. Aldous, Subspaces of $L\sp1$, via random measures, Trans. Amer. Math. Soc. 267 (1981), 445--463.
  • H. Araki, On an inequality of Lieb and Thirring, Lett. Math. Phys. 19 (1990), 167--170.
  • J. Bergh and J. LöFströM, Interpolation Spaces: An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976.
  • E. Berkson, T. A. Gillespie, and P. S. Muhly, Abstract spectral decompositions guaranteed by the Hilbert transform, Proc. London Math. Soc. (3) 53 (1986), 489--517.
  • J. Bourgain, ``Vector-valued singular integrals and the $H\sp 1$-BMO duality'' in Probability Theory and Harmonic Analysis (Cleveland, Ohio, 1983), Monogr. Textbooks Pure Appl. Math. 98, Dekker, New York, 1986, 1--19.
  • B. Erdogan, personal communication, 2006.
  • T. Fack, Type and cotype inequalities for noncommutative $L\sp p$-spaces, J. Operator Theory 17 (1987), 255--279.
  • T. Fack and H. Kosaki, Generalized $s$-numbers of $\tau$-measurable operators, Pacific J. Math. 123 (1986), 269--300.
  • I. C. Gohberg and M. G. KreĭN, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr. 18, Amer. Math. Soc., Providence, 1969.
  • U. Haagerup, ``$L\spp$-spaces associated with an arbitrary von Neumann algebra'' in Algèbres d'opérateurs et leurs applications en physique mathématique (Marseille, 1977), Colloq. Internat. CNRS 274, CNRS, Paris, 1979, 175--184.
  • U. Haagerup, M. Junge, and Q. Xu, Reduction for noncommutative $L_p$-spaces and applications, in preparation.
  • M. Hilsum, Les espaces $L\spp$ d'une algèbre de von Neumann définies par la derivée spatiale, J. Funct. Anal. 40 (1981), 151--169.
  • M. Junge, Doob's inequality for non-commutative martingales, J. Reine Angew. Math. 549 (2002), 149--190.
  • —, Fubini's theorem for ultraproducts of noncommutative $L\sb p$-spaces, Canad. J. Math. 56 (2004), 983--1021.
  • M. Junge, C. Le Merdy, and Q. Xu, Calcul fonctionnel et fonctions carrées dans les espaces $L\sp p$ non commutatifs, C. R. Math. Acad. Sci. Paris 337 (2003), 93--98.
  • M. Junge and J. Parcet, Mixed-norm inequalities and operator space $L_p$ embedding theory, preprint, 2007.
  • M. Junge and H. Rosenthal, Noncommutative $L_p$ spaces are asymptotially stable, in preparation.
  • M. Junge and Q. Xu, Noncommutative Burkholder/Rosenthal inequalities, Ann. Probab. 31 (2003), 948--995.
  • —, Noncommutative maximal ergodic theorems, J. Amer. Math. Soc. 20 (2007), 385--439.
  • V. Kaftal, D. Larson, and G. Weiss, Quasitriangular subalgebras of semifinite von Neumann algebras are closed, J. Funct. Anal. 107 (1992), 387--401.
  • M. I. Kadec and A. PełCzyńSki, Bases, lacunary sequences and complemented subspaces in the spaces $L\sbp$, Studia Math. 21 (1961/1962), 161--176.
  • R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. 1: Elementary Theory, reprint of the 1983 original, Grad. Stud. Math. 15, Amer. Math. Soc., Providence, 1997.
  • —, Fundamentals of the Theory of Operator Algebras, Vol. 2: Advanced Theory, corrected reprint of the 1986 original, Grad. Stud. Math. 16, Amer. Math. Soc., Providence, 1997.
  • H. Kosaki, Applications of the complex interpolation method to a von Neumann algebra: Noncommutative $L\spp$-spaces, J. Funct. Anal. 56 (1984), 29--78.
  • —, Applications of uniform convexity of noncommutative $L\spp$-spaces, Trans. Amer. Math. Soc. 283 (1984), 265--282.
  • —, An inequality of Araki-Lieb-Thirring (von Neumann algebra case), Proc. Amer. Math. Soc. 114 (1992), 477--481.
  • J. L. Krivine and B. Maurey, Espaces de Banach stables, Israel J. Math. 39 (1981), 273--295.
  • S. Kwapień and A. PełCzyńSki, The main triangle projection in matrix spaces and its applications, Studia Math. 34 (1970), 43--68.
  • F. Lust-Piquard, Inégalités de Khintchine dans $C\sb p\;(1<p<\infty)$, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), 289--292.
  • —, A Grothendieck factorization theorem on $2$-convex Schatten spaces, Israel J. Math. 79 (1992), 331--365.
  • F. Lust-Piquard and Q. Xu, The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators, J. Funct. Anal. 244 (2007), 488--503.
  • J. L. Marcolino Nhany, La stabilité des espaces $L\sp p$ non-commutatifs, Math. Scand. 81 (1997), 212--218.
  • V. D. Milman and G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer, Berlin, 1986.
  • E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974), 103--116.
  • G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., Providence, 1986.
  • —, Factorization of operators through $L\sb p\infty$ or $L\sb p1$ and noncommutative generalizations, Math. Ann. 276 (1986), 105--136.
  • —, ``Probabilistic methods in the geometry of Banach spaces'' in Probability and Analysis (Varenna, Italy, 1985), Lecture Notes in Math. 1206, Springer, Berlin, 1986, 167--241.
  • —, Interpolation between $H^p$ spaces and noncommutative generalizations, I, Pacific J. Math. 155 (1992), 341--368.
  • —, Interpolation between $H^p$ spaces and noncommutative generalizations, II, Rev. Mat. Iberoamericana 9 (1993), 281--291.
  • —, The operator Hilbert space OH and type III von Neumann algebras, Bull. London Math. Soc. 36 (2004), 455--459.
  • G. Pisier and Q. Xu, Non-commutative martingale inequalities, Comm. Math. Phys. 189 (1997), 667--698.
  • —, ``Non-commutative $L\sp p$-spaces'' in Handbook of the Geometry of Banach Spaces, Vol. 2, North-Holland, Amsterdam, 2003, 1459--1517.
  • N. Randrianantoanina, Kadec-Pełczyński decomposition for Haagerup $L\sp p$-spaces, Math. Proc. Cambridge Philos. Soc. 132 (2002), 137--154.
  • Y. Raynaud, On ultrapowers of non commutative $L\sb p$ spaces, J. Operator Theory 48 (2002), 41--68.
  • Y. Raynaud and Q. Xu, On subspaces of non-commutative $L\sb p$-spaces, J. Funct. Anal. 203 (2003), 149--196.
  • H. P. Rosenthal, On subspaces of $L\spp$, Ann. of Math. (2) 97 (1973), 344--373.
  • M. Takesaki, Theory of Operator Algebras, I, Springer, New York, 1979.
  • —, Theory of Operator Algebras, II, Encyclopaedia Math. Sci. 125, Oper. Alg. Non-commut. Geom. 6, Springer, Berlin, 2003.
  • —, Theory of Operator Algebras, III, Encyclopaedia Math. Sci. 127, Oper. Alg. Non-commut. Geom. 8, Springer, Berlin, 2003.
  • M. Terp, Interpolation spaces between a von Neumann algebra and its predual, J. Operator Theory 8 (1982), 327--360.
  • —, $L_p$-spaces associated with von Neumann algebras, unpublished manuscript, Institute for Mathematical Sciences, Univ. Copenhagen, Copenhagen, Denmark, 1981.
  • N. Tomczak-Jaegermann, Uniform convexity of unitary ideals, Israel J. Math. 48 (1984), 249--254.