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