Finite rank approximation and semidiscreteness for linear operators

Christian Le Merdy

Annales de l'institut Fourier (1999)

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

Abstract

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Given a completely bounded map u : Z → 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 ⊗ u is bounded from A ⊗ min Z into A ⊗ 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 ⇔ 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.

How to cite

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Merdy, 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

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