Embeddings of non-commutative L p -spaces into preduals of finite von Neumann algebras
Tóm tắt
Let
$$\mathcal{R}$$
be a (not necessarily semi-finite) σ-finite von Neumann algebra. We prove that there exists a finite von Neumann algebra
$$\mathcal{N}$$
so that for every 1 < p < 2, the Haagerup L
p
-space associated with
$$\mathcal{R}$$
embeds isomorphically into
$$\mathcal{N}_ * $$
. We also provide a proof of the following non-commutative generalization of a classical result of Rosenthal: if
$$\mathcal{M}$$
is a semi-finite von Neumann algebra then every reflexive subspace of
$$\mathcal{M}_ * $$
embeds isomorphically into L
r
(
$$\mathcal{M}$$
) for some r > 1.
Tài liệu tham khảo
H. Araki and T. Masuda, Positive cones and L p -spaces for von Neumann algebras, Kyoto University, Research Institute for Mathematical Sciences Publications 18 (1982), 339–411.
C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press Inc., Boston, MA, 1988.
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.
J. Bretagnolle, D. Dacunha-Castelle and J. L. Krivine, Lois stables et espaces L p, Annales de l’Institut Henri Poincaré Sect. B (N.S.) 2 (1965/1966), 231–259.
P. G. Dodds, T. K. Dodds and B. de Pagter, Noncommutative Banach function spaces, Mathematische Zeitschrift 201 (1989), 583–597.
P. G. Dodds, T. K. Dodds and B. de Pagter, Noncommutative Köthe duality, Transactions of the American Mathematical Society 339 (1993), 717–750.
P. G. Dodds, T. K. Dodds, P. N. Dowling, C. J. Lennard, and F. A. Sukochev, A uniform Kadec-Klee property for symmetric operator spaces, Mathematical Proceedings of the Cambridge Philosophical Society 118 (1995), 487–502.
T. Fack, Type and cotype inequalities for noncommutative L p-spaces, Journal of Operator Theory 17 (1987), 255–279.
T. Fack and H. Kosaki, Generalized s-numbers of τ-measurable operators, Pacific Journal of Mathematics 123 (1986), 269–300.
Y. Friedman, Subspace of LC(H) and CC CpC , Proceedings of the American Mathematical Society 53 (1975), 117–122.
U. Haagerup, L p-spaces associated with an arbitrary von Neumann algebra, in Algèbres d’opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977), CNRS, Paris, 1979, pp. 175–184.
U. Haagerup, Noncommutative Integration Theory, Lecture given at the Symposium in Pure Mathematics of the American Mathematical Society, Queens University, Kingston, Ontario, 1980.
U. Haagerup, H. P. Rosenthal and F. A. Sukochev, Banach embedding properties of noncommutative L p-spaces, Memoirs of the American Mathematical Society 163 (2003), no. 776, vi+68.
M. Hilsum, Les éspaces L p d’une algèbre de von Neumann définies par la derivée spatiale, Journal of Functional Analysis 40 (1981), 151–169.
H. Izumi, Constructions of noncommutative L p-spaces with a complex parameter araising from modular actions, International Journal of Mathematics 8 (1997), 1029–1066.
H. Jarchow, On weakly compact operators on C*-algebras, Mathematische Annalen 273 (1986), 341–343.
M. Junge, Noncommutative Poisson random measures, preprint 2005.
M. Junge, Embeddings of noncommutative L p -spaces into noncommutative L 1 -spaces, 1 < p < 2, Geometric and Functional Analysis 10 (2000), 389–406.
M. Junge and J. Parcet, Rosenthal’s theorem for subspaces of noncommutative L p , Duke Mathematical Journal 141 (2008), 75–122.
K. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. I, Academic Press, New York, 1983, Elementary theory.
R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. II, Academic Press, Orlando, FL, 1986, Advanced theory.
H. Kosaki, Applications of the complex interpolation method to a von Neumann algebra: noncommutative L p-spaces, Journal of Functional Analysis 56 (1984), 29–78.
S. Kwapień, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients, Studia Mathematica 44 (1972), 583–595.
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II, Springer-Verlag, Berlin, 1979, Function spaces.
B. Maurey and G. Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Mathematica 58 (1976), 45–90.
E. Nelson, Notes on noncommutative integration, Journal of Functional Analysis 15 (1974), 103–116.
T. Oikhberg, H. P. Rosenthal, and E. Størmer, A Predual Characterization of Semi-finite von Neumann Algebras, in Advances in Quantum Dynamics (South Hadley, MA, 2002), Contemp. Math., vol. 335, Amer. Math. Soc., Providence, RI, 2003, pp. 243–245.
G. Pisier, Factorization of operators through L p∞ or L p1 and noncommutative generalizations, Mathematische Annalen 276 (1986), 105–136.
G. Pisier and Q. Xu, Noncommutative L p-spaces, in Handbook of the Geometry of Banach Spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1459–1517.
N. Randrianantoanina, Embeddings of l p into noncommutative spaces, Journal of the Australian Mathematical Society 74 (2003), 331–350.
Y. Raynaud, L p -spaces associated with a von Neumann algebra without trace: a gentle introduction via complex interpolation, in Trends in Banach Spaces and Operator Theory (Memphis, TN, 2001), Contemp. Math., vol. 321, Amer. Math. Soc., Providence, RI, 2003, pp. 245–273.
Y. Raynaud and Q. Xu, On subspaces of noncommutative L p -spaces, Journal of Functional Analysis 203 (2003), 149–196.
H. P. Rosenthal, On subspaces of L p, Annals of Mathematics (2) 97 (1973), 344–373.
S. Sakai, C*-Algebras and W*-Algebras, Springer-Verlag, New York, 1971.
F. A. Sukochev, Non-isomorphism of L p -spaces associated with finite and infinite von Neumann algebras, Proceedings og the American Mathematical Society 124 (1996), 1517–1527.
F. A. Sukochev and Q. Xu, Embedding of noncommutative L p-spaces: p < 1, Archiv der Mathematik (Basel) 80 (2003), 151–164.
M. Takesaki, Theory of Operator Algebras. I, Springer-Verlag, New York, 1979.
M. Terp, L p-Spaces Associated with von Neumann Algebras, Notes, Copenhagen University, 1981.
P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Press, Cambridge, 1991.
Q. Xu, Embedding of C q and R q into noncommutative L p -spaces, 1 ≤ p ≤ q ≤ 2, Mathematische Annalen 335 (2006), 109–131.
Q. Xu, Analytic functions with values in lattices and symmetric spaces of measurable operators, Mathematical Proceedings of the Cambridge Philosophical Society 109 (1991), 541–563.