Order convergence in ordered vector spaces: compactness, atoms, pervasiveness, and ice cream spaces

Abstract

We study order convergence in ordered vector spaces that are not necessarily lattices. In this setting, we consider two generalizations of order convergence on vector lattices, a directed and an undirected variant. We show that these notions agree for the class of pervasive spaces and extend several results concerning atomicity and order convergence in vector lattices to this class. In particular, we prove that a pervasive JB-algebra is associative, which strengthens a result of Sherman for C*-algebras. We also establish a number of new characterizations of pervasive spaces. Additionally, we systematically study a class of order unit spaces associated to Banach spaces, called ice cream spaces. We provide descriptions of both directed and undirected order convergences in these spaces, yielding examples where the two notions differ. We also consider other order-theoretic aspects of ice cream spaces, such as atoms and lattice operations, relating them to the geometry of the underlying Banach spaces. We demonstrate that ice cream spaces exhibit behavior very different from vector lattices and can serve as a rich source of counterexamples in the theory of ordered vector spaces.