Factorization of vector measures and their integration operators (Q2804280)

The paper is devoted to the study of some generation properties of the space \(L^1(\nu)\) of a (countably additive) vector measure \(\nu\) on a \(\sigma\)-algebra, centering the attention on the connection with uniform Eberlein compacta. The author shows a natural framework for proving known results as the fact that \(L^1(\nu)\) is weakly compactly generated and that it can be isomorphically embedded in a Hilbert generated Banach space. Using the Davis-Figiel-Johnson-Pelczynski factorization theorem and taking into account that each order continuous Banach lattice with a weak unit is weakly compactly generated, we can find a reflexive Banach space and an operator from it of dense range in \(L^1(\nu)\). This is the starting point of a fruitful analysis that leads the author to find several interesting results on the structure of \(L^1(\nu)\) and the integration operator \(I_\nu\).NEWLINENEWLINEThus, for example, Theorem 2.2 shows that every weakly compact set of \(L^1(\nu)\) is uniformly Eberlein compact (UEC), and \((B_{L^1(\nu)^*}, w*)\) is so, which gives a new proof of the fact that \(L^1(\nu)\) is a subspace of a Hilbert generated space. In Section 3, the Davis-Figiel-Johnson-Pelczynski factorization and the arguments used in the previous section, together with some typical factorization tools related to vector measures, are used to provide new relevant information on the integration map \(I_\nu\). The main result of this part is Theorem 3.3, which establishes that, if \(I_\nu\) is completely continuous and Asplund, then \(\nu\) has finite variation \(|\nu|\) and so \(L^1(\nu)=L^1(|\nu|)\).NEWLINENEWLINEThe paper is original and brings together new ideas and classical techniques to get new and highly specialized results on the structure of the spaces \(L^1(\nu)\), which indeed reflects the maturity of the Bartle-Dunford-Schwartz integration theory.