Inventiones mathematicae

, 179:75

Representation of certain homogeneous Hilbertian operator spaces and applications

Article

Abstract

Following Grothendieck’s characterization of Hilbert spaces we consider operator spaces F such that both F and F* completely embed into the dual of a C*-algebra. Due to Haagerup/Musat’s improved version of Pisier/Shlyakhtenko’s Grothendieck inequality for operator spaces, these spaces are quotients of subspaces of the direct sum CR of the column and row spaces (the corresponding class being denoted by QS(CR)). We first prove a representation theorem for homogeneous FQS(CR) starting from the fundamental sequences
$$\Phi _{c}(n)=\Bigg\|\sum_{k=1}^ne_{k1}\otimes e_k\Bigg\|_{C\otimes _{\min}F}^2\quad\mbox{and}\quad \Phi _{r}(n)=\Bigg\|\sum_{k=1}^ne_{1k}\otimes e_k\Bigg\|_{R\otimes _{\min}F}^2$$
given by an orthonormal basis (ek) of F. Under a mild regularity assumption on these sequences we show that they completely determine the operator space structure of F and find a canonical representation of this important class of homogeneous Hilbertian operator spaces in terms of weighted row and column spaces. This canonical representation allows us to get an explicit formula for the exactness constant of an n-dimensional subspace Fn of F:
$$\mathit{ex}(F_n)\sim\biggl[\frac{n}{ \Phi _{c}(n)}\Phi _{r}\bigg(\frac{ \Phi _{c}(n)}{\Phi _{r}(n)}\bigg)+\frac{n}{ \Phi _{r}(n)}\Phi _{c}\bigg(\frac{ \Phi _{r}(n)}{\Phi _{c}(n)}\bigg)\biggr]^{1/2}.$$
In the same way, the projection (=injectivity) constant of Fn is explicitly expressed in terms of Φc and Φr too. Orlicz space techniques play a crucial role in our arguments. They also permit us to determine the completely 1-summing maps in Effros and Ruan’s sense between two homogeneous spaces E and F in QS(CR). The resulting space Π1o(E, F) isomorphically coincides with a Schatten-Orlicz class Sφ. Moreover, the underlying Orlicz function φ is uniquely determined by the fundamental sequences of E and F. In particular, applying these results to the column subspace Cp of the Schatten p-class, we find the projection and exactness constants of Cpn, and determine the completely 1-summing maps from Cp to Cq for any 1≤p, q≤∞.

46L07 47L25

References

1. 1.
Bourgain, J.: Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat. 21, 163–168 (1983)
2. 2.
Burkholder, D.L.: A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional. Ann. Probab. 9, 997–1011 (1981)
3. 3.
Burkholder, D.L.: A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. In: Conference on Harmonic Analysis in Honor of Antoni Zygmund, Wadsworth Math. Ser., vols. I, II, Chicago, Ill., 1981, pp. 270–286. Wadsworth, Belmont (1983) Google Scholar
4. 4.
Effros, Ed., Ruan, Z.-J.: The Grothendieck-Pietsch and Dvoretzky-Rogers theorems for operator spaces. J. Funct. Anal. 122, 428–450 (1994)
5. 5.
Effros, Ed., Ruan, Z.-J.: Operator Spaces. Clarendon Press/Oxford University Press, New York (2000)
6. 6.
Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. American Mathematical Society, Providence (1969)
7. 7.
Haagerup, U., Musat, M.: On the best constants in noncommutative Khintchine-type inequalities. J. Funct. Anal. 250, 588–624 (2007)
8. 8.
Haagerup, U., Musat, M.: The Effros-Ruan conjecture for bilinear forms on C*-algebras. Invent. Math. 174, 139–163 (2008)
9. 9.
Junge, M.: Factorization Theory for Spaces of Operators. Habilitationsschrift, Kiel (1996) Google Scholar
10. 10.
Junge, M.: Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’. Invent. Math. 161, 225–286 (2005)
11. 11.
Junge, M.: Operator spaces and Araki-Woods factors: a quantum probabilistic approach. IMRP Int. Math. Res. Pap., pp. Art. ID 76978, 87 (2006) Google Scholar
12. 12.
Kirchberg, E.: On nonsemisplit extensions, tensor products and exactness of group C *-algebras. Invent. Math. 112, 449–489 (1993)
13. 13.
Kirchberg, E.: Exact C *-algebras, tensor products, and the classification of purely infinite algebras. In: Proceedings of the International Congress of Mathematicians, vols. 1, 2 Zürich, 1994, pp. 943–954. Birkhäuser, Basel (1995) Google Scholar
14. 14.
Kirchberg, E.: On subalgebras of the CAR-algebra. J. Funct. Anal. 129, 35–63 (1995)
15. 15.
Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. I. Springer, Berlin (1977)
16. 16.
Oikhberg, T., Ricard, É: Operator spaces with few completely bounded maps. Math. Ann. 328, 229–259 (2004)
17. 17.
Pisier, G.: Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conference Series in Mathematics, vol. 60. Conference Board of the Mathematical Sciences, Washington (1986) Google Scholar
18. 18.
Pisier, G.: Exact operator spaces. Astérisque 232, 159–186 (1995). Recent advances in operator algebras (Orléans, 1992)
19. 19.
Pisier, G.: The operator Hilbert space OH, complex interpolation and tensor norms. Mem. Am. Math. Soc. 122(585), viii+103 (1996)
20. 20.
Pisier, G.: Non-commutative vector valued L p-spaces and completely p-summing maps. Astérisque 1(247), vi+131 (1998)
21. 21.
Pisier, G.: Introduction to Operator Space Theory. Cambridge University Press, Cambridge (2003)
22. 22.
Pisier, G.: Completely bounded maps into certain Hilbertian operator spaces. Int. Math. Res. Not. 1(74), 3983–4018 (2004)
23. 23.
Pisier, G., Shlyakhtenko, D.: Grothendieck’s theorem for operator spaces. Invent. Math. 150, 185–217 (2002)
24. 24.
Simon, B.: Trace Ideals and Their Applications. Cambridge University Press, Cambridge (1979)
25. 25.
Xu, Q.: Embedding of C q and R q into noncommutative L p-spaces, 1≤p<q≤2. Math. Ann. 335, 109–131 (2006)
26. 26.
Yew, K.-L.: Completely p-summing maps on the operator Hilbert space OH. J. Funct. Anal. 255, 1362–1402 (2008)