## Abstract

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(M) of τ-measurable operators, affiliated with an arbitrary semi-finite von Neumann algebra M equipped with a distinguished faithful, normal, semi-finite trace τ, 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 (ϱ_{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.

Original language | English |
---|---|

Pages (from-to) | 487-502 |

Number of pages | 16 |

Journal | Mathematical Proceedings of The Cambridge Philosophical Society |

Volume | 118 |

Issue number | 3 |

DOIs | |

Publication status | Published - Nov 1995 |

### Bibliographical note

Copyright:Copyright 2019 Elsevier B.V., All rights reserved.