Uniform Kadec-Klee Property in Banach lattices

Abstract

We prove that a Banach lattice $X$ which does not contain the $l^n_{\infty}$-uniformly has an equivalent norm which is uniformly Kadec-Klee for a natural topology $τ$ on $X$. In case the Banach lattice is purely atomic, the topology $τ$ is the coordinatewise convergence topology.