Topologies and convergence structures on vector lattices of operators

This thesis consists of three papers that are centered around the common theme of Hausdorff uo-Lebesgue topologies and convergence structures on vector lattices and on vector lattices and vector lattice algebras of order bounded operators.Its origins lie in asking for possible analogues of the von Neumann bicommutant theorem in the context of Banach lattices and vector lattices. Apart from being interesting in their own right, such analogues are expected to be relevant for the study of vector lattice algebras and Banach lattice algebras of order bounded operators, as well as for representation theory in vector lattices and Banach lattices.