Exact operator spaces

Exact operator spaces

  • Année : 1995
  • Tome : 232
  • Format : Papier, Électronique
  • Langue de l'ouvrage :
  • 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 $d_{SK} (E) = inf \{\Vert u\Vert_{cb} \Vert u^{-1}\Vert _{cb}\}$ where the infimum runs over all isomorphims $u$ between $E$ and an arbitrary $n$-dimensional subspace of the algebra of all compact operators on $l_2$. An operator space $X$ is exact iff $d_{SK} (E)$ remains bounded when $E$ runs over all possible finite dimensional subspaces of $X$. In the general case, it can be shown that $d_{SK} (E) \leq \sqrt n$ (here again $n = dim(E))$, and we give examples showing that this cannot be improved at least asymptotically. We show that $d_{SK} (E) \leq C$ iff for all ultraproducts $\widehat F = \Pi F_i/ {\cal U}$ (of operator spaces) the canonical isomorphism (which has norm $\leq 1) v_{E} : \Pi (E \otimes _{min} F_i) / {\cal U} \rightarrow E \otimes _{min} (\Pi F_i/ {\cal U})$ satisfies $\Vert v^{-1}\Vert \leq C$. Finally, we show that $d_{SK} (E) = d_{SK} (E^{*}) = 1$ holds iff E is a point of continuity with respect to two natural topologies on the set of all $n$-dimensional operator spaces.

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