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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].

46L05 General theory of \(C^*\)-algebras
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
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