Associate space with respect to a semi-finite measure (Q2400602)

The authors study the Köthe dual -- or associate space -- of Banach function spaces over measures that are not in general \(\sigma\)-finite. Although the classical results on abstract Banach lattices and Banach function spaces over \(\sigma\)-finite measures provide a clear picture of the theory in the usual cases, there is a still rather ``unexplored region'' that concerns concrete representations of the dual and the Köthe dual spaces under more general assumptions on the underlying measures. This interesting paper provides some results in this direction. Consider an arbitrary measure space \((\Omega, \Sigma,\mu)\). If \(E\) is a Banach function space over it and \(E^*\) its topological dual, consider the subspace \(E^*_c\) of \(E^*\) of the \(\sigma\)-order continuous functionals and the subspace \(E^*_n\) of the order continuous functionals. Let us write \(E^\times\) for the associate space. Well-known results allow to find the following identification for general Banach lattices: \(E^*=E^*_c\) if \(E\) is \(\sigma\)-order continuous, and \(E^*=E^*_n\) if \(E\) is order continuous. Write now \(\Sigma^f\) for the class of sets in \(\Sigma\) that have finite measure. The measure \(\mu\) is semi-finite if, whenever \(A \in \Sigma\) and \(\mu(A) > 0\), there exists \(B \in \Sigma^f\) with \(B \subset A\) and \(\mu(B)>0\). The measure \(\mu\) is localizable if it is semi-finite and for every collection \(\mathcal A \subset \Sigma\), there is a set \(E \subset \Omega\) such that: {\parindent=0.7cm\begin{itemize}\item[(i)] \(E \in \Sigma\) and \(A \setminus E\) is \(\mu\)-null for each \(A \in \mathcal A\), and \item[(ii)] if \(D \in \Sigma\) and \(A \setminus D\) is \(\mu\)-null for each \(A \in \mathcal A\), then \(E \setminus D\) is null. \end{itemize}} The measure \(\mu\) is locally determined if it is semi-finite and \(\Sigma\) coincides with the set of all \(A \subset \Omega\) such that \( A \cap B \in \Sigma\) for all \(B \in \Sigma^f.\) Each \(\sigma\)-finite measure is localizable, but not conversely. In general, we have that \(E^\times=E^*_n=E^*_c\) when \(\mu\) is \(\sigma\)-finite, and \(E^\times=E^*_n\) when \(\mu\) is localizable. The results of the present paper provide more relations among the given spaces, under some of the assumptions on the measure given above and lattice properties of the Banach function spaces, as the order continuity or the Fatou property. Theorem 3.2 establishes that, for a saturated Banach function space \(E\) and if the measure \(\mu\) is semi-finite, then \(E^\times= E^*_n\) if, and only if, \(E^\times\) has the Fatou property. Corollary 3.3 gives that, if \(\mu\) is semi-finite and \(E\) is a saturated \(\sigma\)-order continuous Banach function space over \(\mu\), then \(E^\times= E^*\) if, and only if, \(E^\times\) has the Fatou property. By Proposition 3.4, we know that, if \(\mu\) is localizable, then \(E^\times\) has the Fatou property; Corollary 3.5 asserts that, under the same assumption for the measure, then \(E^\times=E^*\) if and only if \(E\) is \(\sigma\)-order continuous. Finally, the authors provide in Section 4 an example that shows that, if \(\mu\) is not localizable, the previous results may not be true: the equality \(E^\times = E^*\) can fail for an order continuous Banach function space \(E\) over \(\mu\).