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Central subspaces of Banach spaces. (English) Zbl 0965.46005

Let \(X\) be a Banach space. A subspace \(Y \subset X\) is called a central subspace (\(C\)-subspace) of \(X\) if every finite family of closed balls with centers in \(Y\) that intersects in \(X\) also intersects in \(Y\). In particular \(X\) is said to be generalized center (GC) if and only if \(X\) is a \(C\)-subspace of \(X^{**}\).
Here the authors show that a Banach space \(X\) belongs to the class (GC) if and only if it is a \(C\)-subspace of some dual space. Moreover they show that an \(L^1\)-predual Banach space belongs to the class (GC). For the class of Bochner integrable functions, they obtain that if \(Y\) is a separable \(C\)-subspace of \(X\), then for each \(1 \leq p \leq \infty\), \(L^{p}(\mu, Y)\) is a \(C\)-subspace of \(L^{p}(\mu, X)\) and if moreover \(Y\) is a dual space with the (RNP), then \(L^{p}(\mu, X)\) belongs to the class (GC).

MSC:

46B20 Geometry and structure of normed linear spaces
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
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