The \(L_ 1\) structure of weak \(L_ 1\)
Let \(\mu\) be a nonatomic probability measure. It is shown that the dual space of weak \(L_ 1\) of \(\mu\) contains a number of order isometric copies of \(L_{\infty}(\mu)\). Moreover, the linear span of these copies is \(weak^*\) dense in the dual space. The normed envelope of weak \(L_ 1\) is shown to be covered, in a certain sense, by order isometric, complemented copies of \(L_ 1[0,1]\) (with Lebesgue measure), of \(\ell_ 1[0,1]\), and also, more generally, of the \(\ell_ 1\) direct sum of continuum many disjoint copies of \(L_ 1[0,1]\). But it is not an abstract \(L_ 1\) space, for it also contains order isometric copies of \(\ell_{\infty}\) and of \(c_ 0[0,1]\). Weak \(L_ 1\) itself contains an order isomorphic copy of \(\ell_{\infty}\), and so it is unlikely that the Nikisin factorization of an \(L_ 0\)-valued operator through weak \(L_ 1\) will yield a direct answer to the \(L_ 0\)-factorization problem.
Nikisin factorization of an \(L_ 0\)-valued operator through weak \(L_ 1\)
scientific article; zbMATH DE number 3853777
scientific article