On different modes of order convergence and some applications

TL;DR

This paper reviews various notions of order convergence and their associated topologies, compares them systematically, and applies the findings to von Neumann algebras and measure spaces, revealing new topological equivalences.

Contribution

It provides a comprehensive comparison of different order convergence notions and introduces new results on the topology of von Neumann algebras and $L^ ext{infty}$ spaces.

Findings

Order topology on bounded parts of atomic von Neumann algebras coincides with the $\sigma$-strong topology.

The study extends results to non-$\sigma$-finite von Neumann algebras.

Clarifies relations between order and dual topologies on $L^ ext{infty}$ for semi-finite measure spaces.

Abstract

Different notions for order convergence have been considered by various authors. Associated to every notion of order convergence corresponds a topology, defined by taking as the closed sets those subsets of the poset satisfying that no net in them order converges to a point that is outside of the set. We shall give a thorough overview of these different notions and provide a systematic comparison of the associated topologies. Then, in the last section we shall give an application of this study by giving a result on von Neumann algebras complementing the study started in \cite{ChHaWe}. We show that for every atomic von Neumann algebra (not necessarily $\sigma$-finite) the restriction of the order topology to bounded parts of $M$ coincides with the restriction of the $\sigma$-strong topology $s(M,M_\ast)$. We recall that the methods of \cite{ChHaWe} rest heavily on the assumption of $\sigma$-finiteness. Further to this, for a semi-finite measure space, we shall give a complete picture of the relations between the topologies on $L^\infty$ associated with the duality $\langle L^1, L^\infty\rangle$ and its order topology.