Order topologies on l-algebras

We consider different topologies on an l-algebra associated in a natural way with its order, and briefly study the relationship between them. As a consequence we obtain a partial answer to the problem of the characterization of the topological algebra C(X) of real continuous functions on a topological space X, endowed with its compact convergence topology. In fact, we prove that if A is an Archimedean l-algebra and its order topology τo is locally m-convex, bornological and complete, then there is a realcompact kr-space X such that (A,τo) is isomorphic and homeomorphic with C(X).