Memoirs of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1947-6221(e) ISSN 0065-9266(p)

Mixed-norm inequalities and operator space embedding theory

Author(s): Marius Junge; Javier Parcet
Journal: Memoirs of the AMS 203 (2010), no. 953.
MSC (2000): Primary 46L07, 46L09, 46L51, 46L52, 46L53, 46L54
Posted: August 26, 2009
MathSciNet review: 2589944
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Let $f_1, f_2, \ldots, f_n$ be a family of independent copies of a given random variable in a probability space $(\Omega, \mathcal{F}, \mu)$ . Then, the following equivalence of norms holds whenever $1 \le q \le p < \infty$

$\displaystyle \Big( \int_{\Omega} \Big[ \sum_{k=1}^n \vert f_k\vert^q \Big]^{\f... ...( \int_\Omega \vert f\vert^r d\mu \Big)^{\frac1r} \right\}. \tag{$\Sigma_{pq}$}$

We prove a noncommutative analogue of this inequality for sums of free random variables over a given von Neumann subalgebra. This formulation leads to new classes of noncommutative function spaces which appear in quantum probability as square functions, conditioned square functions and maximal functions. Our main tools are Rosenthal type inequalities for free random variables, noncommutative martingale theory and factorization of operator-valued analytic functions. This allows us to generalize $(\Sigma_{pq})$ as a result for noncommutative

in the category of operator spaces. Moreover, the use of free random variables produces the right formulation of $(\Sigma_{\infty q})$ , which has not a commutative counterpart. In the last part of the paper, we use our mixed-norm inequalities to construct a completely isomorphic embedding of

-equipped with its natural operator space structure- into some sufficiently large

space for $1 \le p < q \le 2$ . The construction of such embedding has been open for quite some time. We also show that hyperfiniteness and the QWEP are preserved in our construction.

References:

1.: H. Araki and E.J. Woods, A classification of factors. Publ. Res. Inst. Math. Sci. Ser. A 4 (1968/69), 51-130. MR 0244773 (39:6087)
2.: J. Bergh and J. Löfström, Interpolation Spaces. Springer-Verlag, Berlin, 1976. MR 0482275 (58:2349)
3.: J. Bretagnolle, D. Dacunha-Castelle and J.L. Krivine, Lois stables et espaces . Ann. Inst. H. Poincaré 2 (1966), 231-259. MR 0203757 (34:3605)
4.: D.L. Burkholder, Distribution function inequalities for martingales. Ann. Probab. 1 (1973), 19-42. MR 0365692 (51:1944)
5.: D.L. Burkholder and R.F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124 (1970), 249-304. MR 0440695 (55:13567)
6.: A. Connes, Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. 6 (1973), 133-252. MR 0341115 (49:5865)
7.: A. Defant and M. Junge, Maximal theorems of Menchoff-Rademacher type in non-commutative -spaces. J. Funct. Anal. 206 (2004), 322-355. MR 2021850 (2005k:46169)
8.: A. Devinatz, The factorization of operator valued analytic functions. Ann. of Math. 73 (1961), 458-495. MR 0126702 (23:A3997)
9.: K. Dykema, Exactness of reduced amalgamated free product $\mathrm{C}^*$ -algebras. Forum Math 16 (2004), 121-189. MR 2039095 (2004m:46142)
10.: E.G. Effros, M. Junge and Z.J. Ruan, Integral mappings and the principle of local reflexivity for noncommutative spaces. Ann. of Math. 151 (2000), 59-92. MR 1745018 (2000m:46120)
11.: E.G. Effros and Z.J. Ruan, Operator Spaces. London Math. Soc. Monogr. 23, Oxford University Press, 2000. MR 1793753 (2002a:46082)
12.: U. Haagerup, Non-commutative integration theory. Unpublished manuscript (1978). See also Haagerup's Lecture given at the Symposium in Pure Mathematics of the Amer. Math. Soc. Queens University, Kingston, Ontario, 1980.
13.: U. Haagerup, spaces associated with an arbitrary von Neumann algebra. Algèbres d'opérateurs et leurs applications en physique mathématique, CNRS (1979), 175-184. MR 560633 (81e:46050)
14.: U. Haagerup, H.P. Rosenthal and F.A. Sukochev, Banach Embedding Properties of Non-commutative -Spaces. Mem. Amer. Math. Soc. 163, 2003. MR 1963854 (2004f:46076)
15.: M. Junge, Embeddings of non-commutative -spaces into non-commutative -spaces, . Geom. Funct. Anal. 10 (2000), 389-406. MR 1771425 (2001f:46098)
16.: M. Junge, Doob's inequality for non-commutative martingales. J. reine angew. Math. 549 (2002), 149-190. MR 1916654 (2003k:46097)
17.: M. Junge, Embedding of the operator space $\mathrm{OH}$ and the logarithmic ‘little Grothendieck inequality’. Invent. Math. 161 (2005), 225-286. MR 2180450 (2006i:47130)
18.: M. Junge, Operator spaces and Araki-Woods factors: A quantum probabilistic approach. Int. Math. Res. Pap. 2006. MR 2268491
19.: M. Junge, Vector-valued spaces over $\mathrm{QWEP}$ von Neumann algebras. In progress.
20.: M. Junge, Noncommutative Poisson random measure. In progress.
21.: M. Junge and M. Musat, A noncommutative version of the John-Nirenberg theorem. Trans. Amer. Math. Soc. 359 (2007), 115-142. MR 2247885 (2007m:46103)
22.: M. Junge and J. Parcet, The norm of sums of independent noncommutative random variables in $L_p(\ell_1)$ . J. Funct. Anal. 221 (2005), 366-406. MR 2124869 (2005k:46151)
23.: M. Junge and J. Parcet, Rosenthal's theorem for subspaces of noncommutatuve . Duke Math. J. 141 (2008), 75-122. MR 2372148
24.: M. Junge and J. Parcet, Operator space embedding of Schatten -classes into von Neumann algebra preduals. Geom. Funct. Anal. 18 (2008), 522-551. MR 2421547
25.: M. Junge and J. Parcet, A transference method in quantum probability. Preprint 2008.
26.: M. Junge, J. Parcet and Q. Xu, Rosenthal type inequalities for free chaos. Ann. Probab. 35 (2007), 1374-1437. MR 2330976
27.: M. Junge and Z.J. Ruan, Decomposable maps on non-commutative spaces. Contemp. Math. 365 (2004), 355-381. MR 2106828 (2005k:46150)
28.: M. Junge and Q. Xu, Noncommutative Burkholder/Rosenthal inequalities. Ann. Probab. 31 (2003), 948-995. MR 1964955 (2004f:46078)
29.: M. Junge and Q. Xu, Noncommutative maximal ergodic theorems. J. Amer. Math. Soc. 20 (2007), 385-439. MR 2276775 (2007k:46109)
30.: M. Junge and Q. Xu, Noncommutative Burkholder/Rosenthal inequalities II: Applications. Israel J. Math. 167 (2008), 227-282. MR 2448025
31.: R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras I and II. Grad. Stud. Math., 15 & 16, American Mathematical Society, 1997. MR 1468229 (98f:46001a); MR 1468230 (98f:46001b)
32.: H. Kosaki, Applications of the complex interpolation method to a von Neumann algebra. J. Funct. Anal. 56 (1984), 29-78. MR 735704 (86a:46085)
33.: S. Kwapień, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. Studia Math. 44 (1972), 583-595. MR 0341039 (49:5789)
34.: E.C. Lance, Hilbert C-modules. Cambridge University Press, 1995. MR 1325694 (96k:46100)
35.: J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I and II. Springer-Verlag, 1996. MR 0500056 (58:17766)
36.: F. Lust-Piquard, Inégalités de Khintchine dans $(1 < p < \infty)$ . C.R. Acad. Sci. Paris 303 (1986), 289-292. MR 859804 (87j:47032)
37.: F. Lust-Piquard and G. Pisier, Non-commutative Khintchine and Paley inequalities. Ark. Mat. 29 (1991), 241-260. MR 1150376 (94b:46011)
38.: M. Musat, Interpolation between non-commutative and non-commutative -spaces. J. Funct. Anal. 202 (2003), 195- 225. MR 1994770 (2004g:46081)
39.: J. Parcet, $\mathrm{B}$ -convex operator spaces. Proc. Edinburgh Math. Soc. 46 (2003), 649-668. MR 2013959 (2004k:46095)
40.: J. Parcet and G. Pisier, Non-commutative Khintchine type inequalities associated with free groups. Indiana Univ. Math. J. 54 (2005), 531-556. MR 2136820 (2006a:46071)
41.: J. Parcet and N. Randrianantoanina, Gundy's decomposition for non-commutative martingales and applications. Proc. London Math. Soc. 93 (2006), 227-252. MR 2235948 (2007b:46115)
42.: G. Pedersen and M. Takesaki, The Radon-Nikodym theorem for von Neumann algebras. Acta Math. 130 (1973), 53-87. MR 0412827 (54:948)
43.: G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces. CBMS (Regional conferences of the A.M.S.) 60, 1987. MR 829919 (88a:47020)
44.: G. Pisier, Projections from a von Neumann algebra onto a subalgebra. Bull. Soc. Math. France 123 (1995), 139-153. MR 1330791 (96f:46111)
45.: G. Pisier, The Operator Hilbert Space OH, Complex Interpolation and Tensor Norms. Mem. Amer. Math. Soc. 122 (1996). MR 1342022 (97a:46024)
46.: G. Pisier, Non-Commutative Vector Valued -Spaces and Completely -Summing Maps. Astérisque 247 (1998).
47.: G. Pisier, Introduction to Operator Space Theory. Cambridge University Press, 2003. MR 2006539 (2004k:46097)
48.: G. Pisier, The operator Hilbert space OH and type III von Neumann algebras. Bull. London Math. Soc. 36 (2004), 455-459. MR 2069007 (2005c:46082)
49.: G. Pisier, Completely bounded maps into certain Hilbertian operator spaces. Internat. Math. Res. Notices 74 (2004), 3983-4018. MR 2103799 (2005g:46114)
50.: G. Pisier and D. Shlyakhtenko, Grothendieck's theorem for operator spaces. Invent. Math. 150 (2002), 185-217. MR 1930886 (2004k:46096)
51.: G. Pisier and Q. Xu, Non-commutative martingale inequalities. Comm. Math. Phys. 189 (1997), 667-698. MR 1482934 (98m:46079)
52.: G. Pisier and Q. Xu, Non-commutative -spaces. Handbook of the Geometry of Banach Spaces II (Ed. W.B. Johnson and J. Lindenstrauss) North-Holland (2003), 1459-1517. MR 1999201 (2004i:46095)
53.: N. Randrianantoanina, Non-commutative martingale transforms. J. Funct. Anal. 194 (2002), 181-212. MR 1929141 (2003m:46098)
54.: N. Randrianantoanina, Conditioned square functions for noncommutative martingales. Ann. Probab. 35 (2007), 1039-1070. MR 2319715 (2009d:46112)
55.: Y. Raynaud, On ultrapowers of non-commutative spaces. J. Operator Theory 48 (2002), 41-68. MR 1926043 (2003i:46069)
56.: H.P. Rosenthal, On the subspaces of spanned by sequences of independent random variables. Israel J. Math. 8 (1970), 273-303. MR 0271721 (42:6602)
57.: H.P. Rosenthal, On subspaces of . Ann. of Math. 97 (1973), 344-373. MR 0312222 (47:784)
58.: Z.J. Ruan, Subspaces of -algebras. J. Funct. Anal. 76 (1988), 217-230. MR 923053 (89h:46082)
59.: D. Shlyakhtenko, Free quasi-free states. Pacific J. Math. 177 (1997), 329-368. MR 1444786 (98b:46086)
60.: M. Takesaki, Tomita's theory of Modular Hilbert Algebras and Its Applications. Lecture Notes in Mathematics 128, Springer, 1970. MR 0270168 (42:5061)
61.: M. Takesaki, Conditional expectations in von Neumann algebras. J. Func. Anal. 9 (1972), 306-321. MR 0303307 (46:2445)
62.: M. Takesaki, Theory of operator algebras I. Springer-Verlag, New York, 1979. MR 548728 (81e:46038)
63.: M. Terp, spaces associated with von Neumann algebras. Math. Institute Copenhagen University, 1981.
64.: M. Terp, Interpolation spaces between a von Neumann algebra and its predual. J. Operator Theory 8 (1982), 327-360. MR 677418 (85b:46075)
65.: D.V. Voiculescu, Symmetries of some reduced free product $\mathrm{C}^*$ -algebras. Operator Algebras and Their Connections with Topology and Ergodic Theory, Springer-Verlag 1132 (1985), 556-588. MR 799593 (87d:46075)
66.: D.V. Voiculescu, A strengthened asymptotic freeness result for random matrices with applications to free entropy. Internat. Math. Res. Notices 1 (1998), 41-63. MR 1601878 (2000d:46080)
67.: D.V. Voiculescu, K. Dykema and A. Nica, Free random variables. CRM Monograph Series 1, American Mathematical Society, 1992. MR 1217253 (94c:46133)
68.: Q. Xu, Recent devepolment on non-commutative martingale inequalities. Functional Space Theory and its Applications. Proceedings of International Conference & 13th Academic Symposium in China. Ed. Research Information Ltd UK. Wuhan 2003, 283-314.
69.: Q. Xu, A description of $(C_p[L_p(M)], R_p[L_p(M)])_{\theta}$ . Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), 1073-1083. MR 2187225 (2006i:47131)
70.: Q. Xu, Operator-space Grothendieck inequalities for noncommutative -spaces. Duke Math. J. 131 (2006), 525-574. MR 2219250 (2007b:46101)
71.: Q. Xu, Embedding of and into noncommutative -spaces, $1 \le p < q \le 2$ . Math. Ann. 335 (2006), 109-131. MR 2217686 (2007m:46095)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|