# Finite rank approximation and semidiscreteness for linear operators

Annales de l'institut Fourier (1999)

- Volume: 49, Issue: 6, page 1869-1901
- ISSN: 0373-0956

## Access Full Article

top## Abstract

top## How to cite

topMerdy, Christian Le. "Finite rank approximation and semidiscreteness for linear operators." Annales de l'institut Fourier 49.6 (1999): 1869-1901. <http://eudml.org/doc/75405>.

@article{Merdy1999,

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.},

author = {Merdy, Christian Le},

journal = {Annales de l'institut Fourier},

keywords = {approximation; tensor products; von Neumann algebras; operators algebras; nuclear -algebras; completely bounded map; nuclearity; semidiscreteness; nets of finite rank operators; tensor map; -semidiscreteness; -nuclearity},

language = {eng},

number = {6},

pages = {1869-1901},

publisher = {Association des Annales de l'Institut Fourier},

title = {Finite rank approximation and semidiscreteness for linear operators},

url = {http://eudml.org/doc/75405},

volume = {49},

year = {1999},

}

TY - JOUR

AU - Merdy, Christian Le

TI - Finite rank approximation and semidiscreteness for linear operators

JO - Annales de l'institut Fourier

PY - 1999

PB - Association des Annales de l'Institut Fourier

VL - 49

IS - 6

SP - 1869

EP - 1901

AB - 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.

LA - eng

KW - approximation; tensor products; von Neumann algebras; operators algebras; nuclear -algebras; completely bounded map; nuclearity; semidiscreteness; nets of finite rank operators; tensor map; -semidiscreteness; -nuclearity

UR - http://eudml.org/doc/75405

ER -

## References

top- [1] R. ARCHBOLD and C. BATTY, C*-norms and slice maps, J. London Math. Soc., 22 (1980), 127-138. Zbl0437.46049MR81j:46090
- [2] D. BLECHER, The standard dual of an operator space, Pacific J. Math., 153 (1992), 15-30. Zbl0726.47030MR93d:47083
- [3] D. BLECHER and V. PAULSEN, Tensor products of operator spaces, J. Funct. Anal., 99 (1991), 262-292. Zbl0786.46056MR93d:46095
- [4] M.-D. CHOI and E. EFFROS, Injectivity and operator spaces, J. Funct. Anal., 24 (1977), 156-209. Zbl0341.46049MR55 #3814
- [5] M.-D. CHOI and E. EFFROS, Nuclear C*-algebras and injectivity: the general case, Indiana Univ. Math. J., 26 (1977), 443-446. Zbl0378.46052MR55 #3799
- [6] M.-D. CHOI and E. EFFROS, Nuclear C*-algebras and the approximation property, Amer. J. Math., 100 (1978), 61-79. Zbl0397.46054MR58 #2317
- [7] E. CHRISTENSEN and A. SINCLAIR, Representations of completely bounded multilinear operators, J. Funct. Anal., 72 (1987), 151-181. Zbl0622.46040MR89f:46113
- [8] A. CONNES, Classification of injective factors, Ann. Math., 104 (1976), 73-115. Zbl0343.46042MR56 #12908
- [9] E. EFFROS and U. HAAGERUP, Lifting problems and local reflexivity for C*-algebras, Duke Math. J., 52 (1985), 103-128. Zbl0613.46047MR86k:46084
- [10] E. EFFROS and A. KISHIMOTO, Module maps and Hochschild-Johnson cohomology, Indiana Univ. Math. J., 36 (1987), 257-276. Zbl0635.46062MR89b:46068
- [11] E. EFFROS and C. LANCE, Tensor products of operator algebras, Adv. Math., 25 (1977), 1-33. Zbl0372.46064MR56 #6402
- [12] E. EFFROS and Z.-J. RUAN, On approximation properties for operator spaces, International J. Math., 1 (1990), 163-187. Zbl0747.46014MR92g:46089
- [13] E. EFFROS and Z.-J. RUAN, On non-self-adjoint operator algebras, Proc. Amer. Soc., 110 (1990), 915-922. Zbl0718.46020MR91c:47086
- [14] E. EFFROS and Z.-J. RUAN, A new approach to operator spaces, Canadian Math. Bull., 34 (1991), 329-337. Zbl0769.46037MR93a:47045
- [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. Zbl0814.47053MR96c:46074a
- [17] U. HAAGERUP, Decomposition of completely bounded maps on operator algebras, unpublished (1980). Zbl0591.46050
- [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. Zbl0591.46050MR87i:46133
- [19] E. HEWITT, The ranges of certain convolution operators, Math. Scand., 15 (1964), 147-155. Zbl0135.36002MR32 #4471
- [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. Zbl0947.46053
- [22] E. KIRCHBERG, C*-nuclearity implies CPAP, Math. Nachr., 76 (1977), 203-212. Zbl0383.46011MR58 #23623
- [23] E. KIRCHBERG, Commutants of unitaries in UHF algebras and functorial properties of exactness, J. Reine Angew. Math., 452 (1994), 39-77. Zbl0796.46043MR95m:46094c
- [24] C. LANCE, On nuclear C*-algebras, J. Funct. Analysis, 12 (1973), 157-176. Zbl0252.46065MR49 #9640
- [25] C. LE MERDY, An operator space characterization of dual operator algebras, Amer. J. Math., 121 (1999), 55-63. Zbl0926.47054MR2001f:46086
- [26] V. PAULSEN, Completely bounded maps on C*-algebras and invariant operator ranges, Proc. Amer. Math. Soc., 86 (1982), 91-96. Zbl0554.46028MR83i:46064
- [27] V. PAULSEN, Completely bounded maps and dilations, Pitman Research Notes in Math., 146, Longman, Wiley, New-York, 1986. Zbl0614.47006MR88h:46111
- [28] V. PAULSEN and S. POWER, Tensor products of non-self-adjoint operator algebras, Rocky Mountain J. Math., 20 (1990), 331-349. Zbl0716.46053MR92d:47061
- [29] V. PAULSEN and R. SMITH, Multilinear maps and tensor norms on operator systems, J. Funct. Anal., 73 (1987), 258-276. Zbl0644.46037MR89m:46099
- [30] G. PISIER, Exact operator spaces, in "Recent Advances in Operator Algebras - Orléans, 1992", Astérisque Soc. Math. France, 232 (1995), 159-187. Zbl0844.46031MR97a:46023
- [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. Zbl0646.46055MR89h:46082
- [33] S. WASSERMANN, Injective C*-algebras, Math. Proc. Cambridge Phil. Soc., 115 (1994), 489-500. Zbl0824.46069
- [34] G. WITTSTOCK, Ein operatorwertiger Hahn-Banach satz, J. Funct. Analysis, 40 (1981), 127-150. Zbl0495.46005MR83d:46072

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.