A uniform Kadec-Klee property for symmetric operator spaces. (English) Zbl 0847.46033
Summary: We show that if a rearrangement invariant Banach function space \(E\) on the positive semi-axis satisfies a non-trivial lower \(q\)-estimate with constant 1 then the corresponding space \(E({\mathcal M})\) of \(\tau\)-measurable operators, affiliated with an arbitrary semifinite von Neumann algebra \(\mathcal M\) equipped with a distinguished faithful, normal, semi-finite trace \(\tau\), has the uniform Kadec-Klee property for the topology of local convergence in measure.
In particular, the Lorentz function spaces \(L_{q, p}\) and the Lorentz-Schatten classes \({\mathcal C}_{q, p}\) have the UKK property for convergence locally in measure and for the weak-operator topology, respectively. As a partial converse, we show that if \(E\) has the UKK property with respect to local convergence in measure then \(E\) must satisfy some non-trivial lower \(q\)-estimate. We also prove a uniform Kadec-Klee result for local convergence in any Banach lattice satisfying a lower \(q\)-estimate.
In particular, the Lorentz function spaces \(L_{q, p}\) and the Lorentz-Schatten classes \({\mathcal C}_{q, p}\) have the UKK property for convergence locally in measure and for the weak-operator topology, respectively. As a partial converse, we show that if \(E\) has the UKK property with respect to local convergence in measure then \(E\) must satisfy some non-trivial lower \(q\)-estimate. We also prove a uniform Kadec-Klee result for local convergence in any Banach lattice satisfying a lower \(q\)-estimate.
MSC:
| 46L51 | Noncommutative measure and integration |
| 46L53 | Noncommutative probability and statistics |
| 46L54 | Free probability and free operator algebras |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Keywords:
space of \(\tau\)-measurable operators; rearrangement invariant Banach function space; semifinite von Neumann algebra; faithful, normal, semi-finite trace; uniform Kadec-Klee property; topology of local convergence in measure; Lorentz function spaces; Lorentz-Schatten classes; UKK property; Banach latticeReferences:
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