SMF

Exact operator spaces

Exact operator spaces

G. PISIER
     
                
  • Année : 1995
  • Tome : 232
  • Format : Électronique
  • Langue de l'ouvrage :
    Anglais
  • Class. Math. : 46L99, 46M99, 46B99
  • Pages : 159-186
  • DOI : 10.24033/ast.317

We study the notion of exactness in the category of operator spaces, in analogy with Kirchberg's work for C-algebras. As for C-algebras, exactness can be characterized either by the exactness of certain sequences, or by the property that the finite dimensional subspaces embed almost completely isometrically into a nuclear C-algebra. Let E be an n-dimensional operator space. We define dSK(E)=inf{ucbu1cb} where the infimum runs over all isomorphims u between E and an arbitrary n-dimensional subspace of the algebra of all compact operators on l2. An operator space X is exact iff dSK(E) remains bounded when E runs over all possible finite dimensional subspaces of X. In the general case, it can be shown that dSK(E)n (here again n=dim(E)), and we give examples showing that this cannot be improved at least asymptotically. We show that dSK(E)C iff for all ultraproducts ˆF=ΠFi/U (of operator spaces) the canonical isomorphism (which has norm 1)vE:Π(EminFi)/UEmin(ΠFi/U) satisfies v1C. Finally, we show that dSK(E)=dSK(E)=1 holds iff E is a point of continuity with respect to two natural topologies on the set of all n-dimensional operator spaces.



Des problèmes avec le téléchargement?
Informez-nous de tout problème que vous avez...