Order closure, order adherence and Fatou norms

Abstract

We record two results in the theory of vector and Banach lattices related to order adherence. First, we give a negative answer to Gao and Leung's question on whether the $uo$-adherence of sublattices must be order closed. Second, we present a self-contained counterexample showing that a Banach lattice with a weakly Fatou norm need not admit any equivalent lattice norm with the Fatou property.