Unbounded Norm Topology in Banach Lattices

Abstract

A net $(x_{\alpha })$ in a Banach lattice $X$ is said to un-converge to a vector $x$ if $\Vert |x_{\alpha }-x|\wedge u\Vert \to 0$ for every $u\in X_{+}$. 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 $X$ is 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.