Local theory of integral Banach mapping spaces
\textit{D. W. Dean} [Proc. Am. Math. Soc. 40, 146--148 (1973; Zbl 0263.46014)] showed that, if \(E\) is a Banach space, then, for any finite-dimensional Banach space \(L\), the equation \(B(L,E)^{**} = B(L,E^{**})\) holds, where \(B(L,E)\) denotes the bounded linear operators from \(L\) to \(E\). He then used this to prove the principle of local reflexivity. This equation in turn served as inspiration for the definition of local reflexivity for operator spaces. Taking inspiration from the theory of operator spaces, the authors study variants of local reflexivity, exactness and injectivity defined by the operator ideal \({\mathcal I}(E,F)\) of integral operators, from a Banach space \(E\) to a Banach space \(F\), with integral norm. In the paper, the term ``Banach mapping space'' is used instead of the more common ``operator ideal'' or ``Banach operator ideal''. A Banach space \(E\) is integral injective if, for any Banach spaces \(F_0 \subseteq F\), every \(\phi_0 \in {\mathcal I}(F_0,E)\) has a linear extension \(\phi \in {\mathcal I}(F,E)\) with the same integral norm. The authors show that every dual Banach space is integral injective. Curiously, the proof of Theorem~2.3 uses a lemma stated for an operator space \(E\), but the proof of Theorem~2.3 should be clear by the duality of injective tensor products, the fact that injective tensor products respect subspaces, and the Hahn-Banach theorem. The authors say that a Banach space is integral locally reflexive if, for any finite-dimensional Banach space \(L\), we have \({\mathcal I}(L,E^{**}) = {\mathcal I}(L,E)^{**}\). It is shown that any Banach space is integral locally reflexive. This is because finite-dimensional Banach spaces are reflexive and have the metric approximation property so that \[ {\mathcal I}(L,E^{**}) = (L \hat{\otimes}_\varepsilon E^*)^* = (L^* \hat{\otimes}_\pi E)^{**} = {\mathcal I}(L,E)^{**}. \] (Here \(\varepsilon\) and \(\pi\), respectively, denote the injective and projective tensor norms.) A Banach space \(E\) can be viewed as a subspace of \(\ell_\infty(I)\) for some set \(I\). Let \(J: E \to \ell_\infty(I)\) be the natural embedding. The authors say that \(E\) is integral exact if there exist integers \(n_\alpha\), linear operators \(\phi_\alpha: E \to \ell_\infty^{n_\alpha}\), and \(\psi_\alpha : \ell_\infty^{n_\alpha} \to \ell_\infty(I)\) all with integral norm less than one such that for all \(x \in E\) \[ \lim_\alpha \|J(x) - \psi_\alpha(\phi_\alpha(x))\| \to 0. \] The main result is that a Banach space \(E\) is integral exact if and only if \(E = \mathbb{C}\).
scientific article; zbMATH DE number 6898331
scientific article