This is an announcement for the paper “Unbounded Norm Topology in Banach Lattices” by M. Kandićhttp://arxiv.org/find/math/1/au:+Kandic_M/0/1/0/all/0/1, M.A.A. Marabehhttp://arxiv.org/find/math/1/au:+Marabeh_M/0/1/0/all/0/1, V.G. Troitskyhttp://arxiv.org/find/math/1/au:+Troitsky_V/0/1/0/all/0/1.
Abstract: A net $(x_{\alpha})$ in a Banach lattice $X$ is said to un-converge to a vector $x$ if $||x_\alpha-x|\wedge u||\rightarrow 0$ 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..
The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1608.05489