A uniform Kadec-Klee property for symmetric operator spaces

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 Lq,p and the Lorentz-Schatten classes Cq,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.