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Banach space-valued extensions of linear operators on \(L^\infty\). (English) Zbl 1415.46027

de Jeu, Marcel (ed.) et al., Ordered structures and applications. Positivity VII (Zaanen centennial conference), Leiden, the Netherlands, July 22–26, 2013. Basel: Birkhäuser/Springer. Trends Math., 281-306 (2016).
Given a Banach space \(X\) and a Banach function space \(E\) on a measure space \((A, {\mathcal{A}}, \mu)\), denote by \(E(X)\) the Köthe-Bochner space associated with \(E\) and \(X\). On the other hand, recall that a Banach dual pair is a triple \((X, Y, \langle \cdot, \cdot \rangle)\), consisting of two Banach spaces \(X\) and \(Y\) with a Banach duality between them, that is, a bounded bilinear form \(\langle \cdot, \cdot \rangle:X \times Y \to \mathbb K\) for which the induced linear maps \(x \mapsto \langle x, \cdot \rangle\), \(X \to Y^{*}\) and \(y \mapsto \langle \cdot, y \rangle\), \(Y \to X^{*}\) are injections.
Let \(T\in {\mathcal L}(E, Y)\), and let \((X, Y, \langle \cdot, \cdot \rangle)\) be a Banach dual pair. In this paper, the author gives conditions under which there exists a (necessarily unique) bounded linear operator \(T_{Y} \in {\mathcal L}(E(Y),G(Y))\) with the property that \[ \langle x, T_{Y} e \rangle= T \langle x, e \rangle \text{ for all }e \in E(Y), \,x \in X. \]
The first main result states that, in the case \(\langle X, Y \rangle=\langle Y^{*}, Y \rangle\), with \(Y\) a reflexive Banach space, for the existence of \(T_{Y}\) it is sufficient that \(T\) is dominated by a positive operator. Then, it is shown that, for \(Y\) within a wide class of Banach spaces (including the Banach lattices), the validity of this extension result for \(E = l_{\infty}\) and \(G=\mathbb K\) even characterizes the reflexivity of \(Y\).
The second main result concerns the case that \(T\) is an adjoint operator on \(L^{\infty}(A)\): it is assumed that \(E = L^{\infty}(A)\) for a semi-finite measure space \((A, {\mathcal A}, \mu)\), that \(F,G\) is a Köthe dual pair, and that \(T\) is \(\sigma(L^{\infty} (A),L^{1}(A))\)-to-\(\sigma(G, F)\) continuous. In this situation, it is proved that \(T_{Y}\) also exists, provided that \(T\) is dominated by a positive operator. As an application of this result, the conditional expectation on Banach space-valued \(L^\infty\)-spaces is considered.
For the entire collection see [Zbl 1354.46003].

MSC:

46E40 Spaces of vector- and operator-valued functions
46B10 Duality and reflexivity in normed linear and Banach spaces
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References:

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