Measures of weak non-compactness in preduals of von Neumann algebras and JBW${}^{\ast }$-triples

Abstract

We prove, among other results, that three standard measures of weak non-compactness coincide in preduals of JBW${}^{\ast }$-triples. This result is new even for preduals of von Neumann algebras. We further provide a characterization of JBW${}^{\ast }$-triples with strongly WCG predual and describe the order of seminorms defining the strong${}^{\ast }$ topology. As a byproduct we improve a characterization of weakly compact subsets of a JBW${}^{\ast }$-triple predual, providing so a proof for a conjecture, open for almost eighteen years, on weakly compact operators from a JB${}^{\ast }$-triple into a complex Banach space.

58 pages; the paper was revised - some misprints were corrected, some proofs were done in more detail, some proofs were simplified, references were updated