Algebras of Uniformly Continuous Functions

Abstract

There is a well-known dual relationship between compact Hausdorff spaces X and the algebras C(X) of all real-valued continuous functions on the spaces. This generalizes, imperfectly of course, to completely regular spaces X; the algebras C(X) are no longer Banach algebras, and it is not too well understood what they are. This paper concerns a slightly more general class of algebras C(pX) corresponding dually to certain' uniform spaces pX. (One obtains C(X) by choosing ,p so that all continuous functions are uniformly continuous.) Most of the known results for unbounded function algebras C(X) go over, and in generalizing we obtain two main advantages. The first main advantage is an analogue of the Stone-Weierstrass theorem. That theorem tells us that the algebras C(X), for X compact, are just the algebras of bounded real-valued functions which are complete under uniform convergence. The present algebras C(pX) are those algebras of real-valued functions which are complete under uniform convergence and closed under all continuous n-ary operations; that is, if Al -,fn are in the algebra and g is a continuous real-valued function on En, then g(f1, *-,ffn) is in the algebra. (Also, any subalgebra closed under all operations and inducing the given uniformity is uniformly dense. No analogous results are known for general C(X).) The second main advantage is an analogue of Tietze's extension theorem: every uniformly continuous real valued function on a subspace of a uniform space, of the special class here considered, can be extended over the whole space. Then by considering only complete spaces one has a perfect bicategorical duality between the spaces fiX and the algebras C(pX), subspaces corresponding to quotient algebras, and quotient spaces to subalgebras. (Categorical duality means a suitable correspondence between algebra homomorphisms and continuous mappings, and the term bicategorical covers the suband quotient-systems. The latter concept was first formalized by MacLane [20], but a slightly different treatment is developed from scratch in this paper.) * The author is a National Science Foundation fellow. The work was advanced by, and reported on to, the 1955 National Science Foundation-American Mathematical Society Summer Institute on Set Theoretic Topology. Defined below. 96