%0 Journal Article
%T A Uniform Kadec-klee Property For Symmetric Operator Spaces
%A Peter G. Dodds
%A T. K. Dodds
%A Paddy N. Dowling
%A Christopher J. Lennard
%A Fyodor A. Sukochev
%J Mathematics
%D 1993
%I arXiv
%X 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 $E(\nm)$ of $\tau-$measurable operators, affiliated with an arbitrary semi-finite von Neumann algebra $\nm$ 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 ${\cal C}_{q,p}$ 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.
%U http://arxiv.org/abs/math/9301201v1