On different modes of order convergence and some applications

Kevin Abela, Emmanuel Chetcuti, Hans Weber

Published 2020-12-26Version 1

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.