Unbounded Norm Topology in Banach Lattices

TL;DR

This paper explores the properties of un-topology in Banach lattices, characterizing when it coincides with norm topology, is metrizable, locally convex, and its relation to un-compactness and weak*-convergence.

Contribution

It provides new characterizations of un-topology properties in Banach lattices, including conditions for metrizability, local convexity, and un-compactness, linking them to lattice structure.

Findings

Un-topology agrees with norm topology iff the lattice has a strong unit.

Un-topology is metrizable iff the lattice has a quasi-interior point.

Un-compactness of the unit ball characterizes atomic KB-spaces.

Abstract

A net $(x_\alpha)$ in a Banach lattice $X$ is said to un-converge to a vector $x$ if $\bigl\lVert\lvert x_\alpha-x\rvert\wedge u\bigr\rVert\to 0$ 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.