The Gelfand-Phillips and Dunford-Pettis type properties in bimodules of measurable operators. (English) Zbl 07905994
Summary: We fully characterize noncommutative symmetric spaces \(E(\mathcal{M},\tau )\) affiliated with a semifinite von Neumann algebra \(\mathcal{M}\) equipped with a faithful normal semifinite trace \(\tau\) on a (not necessarily separable) Hilbert space having the Gelfand-Phillips property and the WCG-property. The complete list of their relations with other classical structural properties (such as the Dunford-Pettis property, the Schur property and their variations) is given in the general setting of noncommutative symmetric spaces.
MSC:
| 46L52 | Noncommutative function spaces |
| 46B03 | Isomorphic theory (including renorming) of Banach spaces |
| 47B07 | Linear operators defined by compactness properties |
| 46L10 | General theory of von Neumann algebras |
| 46B85 | Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science |
Keywords:
noncommutative symmetric space; order continuous norm; Gelfand-Phillips space; WCG-space; (weak/strong) Dunford-Pettis propertyReferences:
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