Some results on non-commutative Banach function spaces
Let \({\mathcal M}\) be a semifinite von Neumann algebra. We define a system of axioms for non-commutative Banach function spaces as an analogue of that for commutative spaces. We show that, if \({\mathcal M}\) is a \(\sigma\)-finite II\(_ 1\)-factor, then there exists a bijective correspondence between rearrangement invariant function norms on \([0,1]\) and non-commutative Banach function spaces associated with \({\mathcal M}\). Furthermore, the separability of those spaces are characterized in terms of the function norms and the predual of \({\mathcal M}\).
scientific article; zbMATH DE number 108645
scientific article