# Measures of weak non-compactness in preduals of von Neumann algebras and JBW$^*$-triples

## TL;DR

This paper establishes the equivalence of three measures of weak non-compactness in preduals of JBW*-triples, providing new characterizations and resolving an 18-year-old conjecture about weakly compact operators.

## Contribution

It proves the coincidence of measures of weak non-compactness in preduals of JBW*-triples and characterizes those with strongly WCG preduals, also resolving a long-standing conjecture.

## Key findings

- Three measures of weak non-compactness coincide in preduals of JBW*-triples.
- Characterization of JBW*-triples with strongly WCG predual.
- Proof of an 18-year-old conjecture on weakly compact operators.

## 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.