Finite rank approximation and semidiscreteness for linear operators

Merdy, Christian Le

Annales de l'Institut Fourier, Tome 49 (1999), p. 1869-1901 / Harvested from Numdam

Résumé

Étant donnée une application complètement bornée $u:Z\rightarrow M$ d’un espace d’opérateurs $Z$ dans une algèbre de von Neumann (ou simplement une algèbre duale unifère) $M$, on dit que $u$ est $C$-semi-discrète si pour toute algèbre d’opérateurs $A$, le produit tensoriel d’opérateurs $I_A\otimes u$ est borné de $A\otimes _{\rm min} Z$ dans $A\otimes _{\rm nor} M$, avec une norme inférieure ou égale à $C$. Nous étudions cette propriété et la caractérisons notamment par des propriétés d’approximation appropriées, qui généralisent la caractérisation des algèbres de von Neumann semi-discrètes due à Choi-Effros. Notre travail est une extension de travaux récents de Pisier sur une notion comparable de $C^*$-nucléarité pour les applications linéaires. Ayant à l’esprit l’équivalence “$B$ est nucléaire $\Leftrightarrow B^{**}$ est semi-discrète” valable pour une $C^*$-algèbre $B$, nous étudions les relations qui existent entre la nucléarité d’une application linéaire et le caractère semi-discret de son biadjoint. Enfin nous obtenons, grâce à certaines de nos techniques, de nouvelles propriétés de la norme de Haagerup pour les opérateurs décomposables entre $C^*$-algèbres.

Abstract

Given a completely bounded map $u:Z\rightarrow M$ from an operator space $Z$ into a von Neumann algebra (or merely a unital dual algebra) $M$, we define $u$ to be $C$-semidiscrete if for any operator algebra $A$, the tensor operator $I_A\otimes u$ is bounded from $A\otimes _{\rm min} Z$ into $A\otimes _{\rm nor} M$, with norm less than $C$. We investigate this property and characterize it by suitable approximation properties, thus generalizing the Choi-Effros characterization of semidiscrete von Neumann algebras. Our work is an extension of some recent work of Pisier on an analogous generalization of $C^*$-nuclearity to operators. Having in mind the equivalence “$B$ is nuclear $\Leftrightarrow B^{**}$ semidiscrete”, when $B$ is a $C^*$-algebra, we then study the relationships between the nuclearity of an operator and the semidiscreteness of its decomposable norm on operators between $C^*$-algebras. Lastly, we apply some of our techniques to find new properties of Haagerup’s decomposable norm on operators between $C^*$-algebras.

Détail

Publié le : 1999-01-01

MR 2001b:46092 | Zbl 0989.46033

DOI : https://doi.org/10.5802/aif.1741

Bibliographie

[1] R. Archbold and C. Batty, C*-norms and slice maps, J. London Math. Soc., 22 (1980), 127-138. | MR 81j:46090 | Zbl 0437.46049

[2] D. Blecher, The standard dual of an operator space, Pacific J. Math., 153 (1992), 15-30. | MR 93d:47083 | Zbl 0726.47030

[3] D. Blecher and V. Paulsen, Tensor products of operator spaces, J. Funct. Anal., 99 (1991), 262-292. | MR 93d:46095 | Zbl 0786.46056

[4] M.-D. Choi and E. Effros, Injectivity and operator spaces, J. Funct. Anal., 24 (1977), 156-209. | MR 55 #3814 | Zbl 0341.46049

[5] M.-D. Choi and E. Effros, Nuclear C*-algebras and injectivity: the general case, Indiana Univ. Math. J., 26 (1977), 443-446. | MR 55 #3799 | Zbl 0378.46052

[6] M.-D. Choi and E. Effros, Nuclear C*-algebras and the approximation property, Amer. J. Math., 100 (1978), 61-79. | MR 58 #2317 | Zbl 0397.46054

[7] E. Christensen and A. Sinclair, Representations of completely bounded multilinear operators, J. Funct. Anal., 72 (1987), 151-181. | MR 89f:46113 | Zbl 0622.46040

[8] A. Connes, Classification of injective factors, Ann. Math., 104 (1976), 73-115. | MR 56 #12908 | Zbl 0343.46042

[9] E. Effros and U. Haagerup, Lifting problems and local reflexivity for C*-algebras, Duke Math. J., 52 (1985), 103-128. | MR 86k:46084 | Zbl 0613.46047

[10] E. Effros and A. Kishimoto, Module maps and Hochschild-Johnson cohomology, Indiana Univ. Math. J., 36 (1987), 257-276. | MR 89b:46068 | Zbl 0635.46062

[11] E. Effros and C. Lance, Tensor products of operator algebras, Adv. Math., 25 (1977), 1-33. | MR 56 #6402 | Zbl 0372.46064

[12] E. Effros and Z.-J. Ruan, On approximation properties for operator spaces, International J. Math., 1 (1990), 163-187. | MR 92g:46089 | Zbl 0747.46014

[13] E. Effros and Z.-J. Ruan, On non-self-adjoint operator algebras, Proc. Amer. Soc., 110 (1990), 915-922. | MR 91c:47086 | Zbl 0718.46020

[14] E. Effros and Z.-J. Ruan, A new approach to operator spaces, Canadian Math. Bull., 34 (1991), 329-337. | MR 93a:47045 | Zbl 0769.46037

[15] E. Effros and Z.-J. Ruan, Operator convolution algebras: an approach to quantum groups, unpublished (1991).

[16] E. Effros and Z.-J. Ruan, Mapping spaces and liftings for operator spaces, Proc. London Math. Soc., 69 (1994), 171-197. | MR 96c:46074a | Zbl 0814.47053

[17] U. Haagerup, Decomposition of completely bounded maps on operator algebras, unpublished (1980). | Zbl 0591.46050

[18] 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 87i:46133 | Zbl 0591.46050

[19] E. Hewitt, The ranges of certain convolution operators, Math. Scand., 15 (1964), 147-155. | MR 32 #4471 | Zbl 0135.36002

[20] M. Junge, Factorization theory for spaces of operators, Habilitationsschrift, Universitat Kiel, 1996.

[21] M. Junge and C. Le Merdy, Factorization through matrix spaces for finite rank operators between C*-algebras, Duke Math. J., to appear. | Zbl 0947.46053

[22] E. Kirchberg, C*-nuclearity implies CPAP, Math. Nachr., 76 (1977), 203-212. | MR 58 #23623 | Zbl 0383.46011

[23] E. Kirchberg, Commutants of unitaries in UHF algebras and functorial properties of exactness, J. Reine Angew. Math., 452 (1994), 39-77. | MR 95m:46094c | Zbl 0796.46043

[24] C. Lance, On nuclear C*-algebras, J. Funct. Analysis, 12 (1973), 157-176. | MR 49 #9640 | Zbl 0252.46065

[25] C. Le Merdy, An operator space characterization of dual operator algebras, Amer. J. Math., 121 (1999), 55-63. | MR 2001f:46086 | Zbl 0926.47054

[26] V. Paulsen, Completely bounded maps on C*-algebras and invariant operator ranges, Proc. Amer. Math. Soc., 86 (1982), 91-96. | MR 83i:46064 | Zbl 0554.46028

[27] V. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Math., 146, Longman, Wiley, New-York, 1986. | MR 88h:46111 | Zbl 0614.47006

[28] V. Paulsen and S. Power, Tensor products of non-self-adjoint operator algebras, Rocky Mountain J. Math., 20 (1990), 331-349. | MR 92d:47061 | Zbl 0716.46053

[29] V. Paulsen and R. Smith, Multilinear maps and tensor norms on operator systems, J. Funct. Anal., 73 (1987), 258-276. | MR 89m:46099 | Zbl 0644.46037

[30] G. Pisier, Exact operator spaces, in "Recent Advances in Operator Algebras - Orléans, 1992", Astérisque Soc. Math. France, 232 (1995), 159-187. | MR 97a:46023 | Zbl 0844.46031

[31] G. Pisier, An introduction to the theory of operator spaces, preprint (1997).

[32] Z.-J. Ruan, Subspaces of C*-algebras, J. Funct. Analysis, 76 (1988), 217-230. | MR 89h:46082 | Zbl 0646.46055

[33] S. Wassermann, Injective C*-algebras, Math. Proc. Cambridge Phil. Soc., 115 (1994), 489-500. | Zbl 0824.46069

[34] G. Wittstock, Ein operatorwertiger Hahn-Banach satz, J. Funct. Analysis, 40 (1981), 127-150. | MR 83d:46072 | Zbl 0495.46005