Exact operator spaces.

*(English)*Zbl 0844.46031
Connes, A. (ed.), Recent advances in operator algebras. Collection of talks given in the conference on operator algebras held in Orléans, France in July 1992. Paris: Société Mathématique de France, Astérisque. 232, 159-186 (1995).

Summary: 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 \(d_{SK} (E)= \inf\{|u|_{cb} |u^{-1} |_{cb}\}\) where the infimum runs over all isomorphisms \(u\) between \(E\) and an arbitrary \(n\)-dimensional subspace of the algebra of all compact operators on \(\ell_2\). 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 \(\widehat {F}= \Pi F_i/{\mathcal U}\) (of operator spaces) the canonical isomorphism (which has norm \(\leq 1\)) \(v_E: \Pi(E \otimes_{\min} F_i)/ {\mathcal U}\to E\otimes_{\min} (\Pi F_i/ {\mathcal U})\) satisfies \(|v_E^{-1} |\leq C\). 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.

For the entire collection see [Zbl 0832.00041].

For the entire collection see [Zbl 0832.00041].