Remarks on certain function spaces

Abstract

Section t . Introduction The norm topology in C(S) of uniform convergence on S (here S is locally compact Hausdorff and C(S) denotes the algebra of complex-valued bounded continuous functions on S) is an important one and has been studied extensively, However, it does not seem the appropriate topology to place on C(S) if one wishes to study C(S) as a topological vector space or algebra, particularly if one wishes to relate properties in C(S) with those in S. In fact, as a topological vector space or algebra, C(S) when given this topology is the same as C~S), where flS is the Stone-Cech compactification of S. A more natural topology on C(S) for S non-compact would be one which satisfies at least the following properties: (1) the continuous multiplicative linear forms on C(S) are given (via evaluation) by points of S, (2) the topology coincides with the uniform topology when S is compact. Such a topology was introduced and studied by Buck [3, 4], who called it the fl or strict topology. Additional results on the space (C(S), fl) have been obtained recently by CONWAY [7, 8, 9, 10], TODD [22], WELLS [24, 25], and COLLINS [5], results which implement to some degree the notion that fl is a natural and useful topology to place on C(S). It is the purpose of this paper to examine further the space (C(S), fl), the work being divided into four sections. In section 3 it is shown that (C(S), fl) always has the approximation property. These results of section 3 include a theorem of GROTHENDIECK [16, I, p. 185] but our method of proof is considerably more direct and revealing. In section 4 two useful theorems are proved concerning special types of approximate identities for the Banach algebra Co(S ) of complex continuous functions which vanish at Do (supremum norm). It is known [4, p. 98] that bounded approximate identities always exist, but it is shown here that a sequential one exists if and only if S is a compact and a fl totally bounded one exists when S is paracompact. Section 5 contains two unrelated but possibly interesting results: (1) if S is paracompact or if (C(S), fl) is a (DF) space then M(S) is a(M, C) sequentially complete (here M(S) is the space of bounded Radon measures on S and tr(M, C) is the weak * topology imposed on M(S) by C(S)), (2) if each uniformly bounded real sequence in Co(S ) has an upper bound in Co(S ) (in particular, if (C(S), fl) is a (DF) space) then S can contain no closed C* embedded copy of N, the natural numbers (and certainly S must be pseudocompact). In the final section (section 6)