Some Applications of the Ultrapower Theorem to the Theory of Compacta

In the ten years since the celebration of Professor Banaschewski’s 60th birthday (and the writing of [4] in commemoration), there has been a fair amount of development in the theory of ultracoproducts of compacta (i.e., compact Hausdorff spaces). Papers [3, 5, 7, 6] have been written by this author; also there is the paper [13] by the late R. Gurevič. In addition to this, there has been a parallel development in the firstorder theory of Banach spaces/algebras begun by C. W. Henson in [16]. There are important links between Banach space theory and our work (mainly through Gel’fandNăımark duality), and we present here two applications of Banach techniques to the theory of compacta. (See 3.1 and 4.2 below.) Since Gel’fand-Năımark duality is a two-way street, much of the “dualized” model theory developed for compacta may be directly translated into the Banach model theory of commutative B-algebras. (See the results in §5 and §6 below.) It seems likely that the future will see much in the way of progress in these two streams of research, as topological issues stimulate the analytic and vice versa. We begin with a quick review of the topological ultracoproduct construction; detailed accounts may be found in [2, 3, 4, 5, 7, 6, 13]. We letCH denote the category of compacta and continuous maps. In model theory, it is well known that ultraproducts (and reduced products in general, but we restrict ourselves to maximal filters on the index set) may be described in the language of category theory; i.e., as direct limits of (cartesian) products, where the directed set is the ultrafilter with reverse inclusion, and the system of products consists of cartesian products taken over the various sets in the ultrafilter. (Bonding maps are just the obvious restriction maps.) When we transport this framework to CH, the result is