On injectivity and nuclearity for operator spaces
Operator spaces can be regarded as quantized Banach spaces and it is interesting to introduce quantized analogues of the theory of Banach spaces. A quantized object being recently investigated is an injective operator space. An operator space \(V\) is said to be injective (or \(1\)-injective, according to \textit{M. Neal} and \textit{B. Russo} [C. R. Acad. Sci., Paris, Sér. I, Math. 331, 873--878 (2000; Zbl 0973.47052)]) if for any operator spaces \(W_1 \subseteq W_2\), each complete contraction \(\varphi:W_1 \rightarrow V\) has a completely contractive extension \(\tilde{\varphi}:W_2 \rightarrow V\). For example, \(B(H)\) is an injective operator space by the celebrated Arveson-Wittstock Hahn-Banach theorem (see Theorem 7.2 of \textit{V. I. Paulsen} [Completely Bounded Maps and Dilations (Pitman Res. Notes Math. Ser. 146), Longman (1986; Zbl 0614.47006)], or Section 3 of \textit{E. G. Effros} and \textit{Z.-J. Ruan} [Pac. J. Math. 132, 243--264 (1988; Zbl 0686.46012)]). Ruan proved that an operator space \(V\) is injective if and only if there is an injective \(C^*\)-algebra \(A\) and a projection \(p \in A\) such that \(V\) is completely isometrically isomorphic to \(pA(1-p)\); see Theorems 4.3 and 4.5 of \textit{Z.-J. Ruan} [Trans. Am. Math. Soc. 315, 89--104 (1989; Zbl 0669.46029)]. The authors prove that an operator space that is dual as a Banach space is injective if and only if it is completely isometric and weak* homeomorphic to \(eR(1-e)\) for some injective von Neumann algebra \(R\) and some projection \(e\in R\). Recall that a von Neumann algebra is called injective if it is the image of a projection of norm \(1\) from \(B(H)\). It is also proved that (isometrically) exact operator spaces are always locally reflexive and that an operator space \(V\) is nuclear if and only if it is locally reflexive and \(V^{**}\) is injective (i.e., semidiscrete). An operator space \(V\) is said to be locally reflexive if the injective operator space tensor product \(F \otimes V^{**}\) coincides with \((F \otimes V)^{**}\) for all finite-dimensional operator spaces \(F\); see \textit{E. G. Effros}, \textit{M. Junge} and \textit{Z.-J. Ruan} [Ann. Math. (2) 151, 59--92 (2000; Zbl 0957.47051)]. The authors, by modifying the operator space matrix norm on a von Neumann algebra, show that there exist operator spaces that are dual Banach spaces but which are not dual operator spaces.
scientific article; zbMATH DE number 1820862
scientific article