Order topologies on \(l\)-algebras.
The order topology and abstractions of the order topology are studied in the setting of lattice-ordered algebras. The order topology is characterized as the topology defined by all lattice seminorms. A seminorm is said be a lattice seminorm if \(q(x)\leq q(y)\) whenever \(| x| \leq| y| \). It is important in the context of algebras to consider m-convex sets or m-seminorms (\(q\) is an m-seminorm if it satisfies \(q(fg)\leq q(f)q(g) \) for \(f\) and \(g\) in the lattice-ordered algebra). Thus m-convex topologies can be considered. This paper provides an instructive comparison of different topologies on lattice-ordered algebras as well as conditions when the order topology is m-convex. In the special case of uniformly closed Archimedean lattice-ordered algebras where \(1\) is a weak order unit, a number of the topologies under consideration coincide. Further, if these algebras are also closed under inversion, then the order topology coincides with an appropriate topology of compact convergence (thus, m-convex). This analysis leads to a characterization of certain spaces \( C_k(X) \), all continuous real-valued functions on \(X\) with the topology of compact convergence . In particular, it is shown that for an \(f\)-algebra \(A\) there exists a topology that is locally m-convex, bornological, complete, and generated by lattice seminorms if and only if \(A\) with its order topology is isomorphic and homeomorphic to \(C_k(X)\) for a realcompact \( k_r \)-space.
scientific article; zbMATH DE number 2056856
scientific article