Unbounded norm topology in Banach lattices

Abstract

A net (xα) in a Banach lattice X is said to un-converge to a vector x if || |xα−x|∧u ||→0 for every u ∈X+. In this paper, we investigate un-topology, i.e., the topology that corresponds to un-convergence. We show that un-topology agrees with the norm topology iff X has a strong unit. Un-topology is metrizable iff X has a quasi-interior point. Suppose that X is order continuous, then un-topology is locally convex iff X is atomic. An order continuous Banach lattice Xis a KB-space iff its closed unit ball B_X is un-complete. For a Banach lattice X, B_X is un-compact iff X is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak*-convergence.