More Smoothly Real Compact Spaces

A topological space X is called A-real compact, if every algebra homomorphism from A to the reals is an evaluation at some point of X, where A is an algebra of continuous functions. Our main interest lies on algebras of smooth functions. In [AdR] it was shown that any separable Banach space is smoothly real compact. Here we generalize this result to a huge class of locally convex spaces including arbitrary products of separable Fréchet spaces. In [KMS] the notion of real compactness was generalized, by defining a topological space X to be A-real-compact, if every algebra homomorphism α : A → R is just the evaluation at some point a ∈ X , where A is a some subalgebra of C(X,R). In case A equals the algebra C(X,R) of all continuous functions this condition reduces to the usual real-compactness. Our main interest lies on algebras A of smooth functions. In particular we showed in [KMS] that every space admitting A-partitions of unity is A-real-compact. Furthermore any product of the real line R is C-real-compact. A question we could not solve was, whether l is C-real-compact, despite the fact that there are no smooth bump functions. [AdR] had already shown that this is true not only for l, but for any separable Banach space. The aim of this paper is to generalize this result of [AdR] to a huge class of locally convex spaces, including arbitrary products of separable Fréchet spaces. Typeset by AMS-TEX 1 2 A. KRIEGL, P. W. MICHOR Convention. All subalgebras A ⊆ C(X,R) are assumed to be real algebras with unit and with the additional property that for any f ∈ A with f(x) 6= 0 for all x ∈ X the function 1 f lies also in A. 1. Lemma. Let A ⊂ C(X,R) be a finitely generated subalgebra of continuous functions on a topological space X. Then X is A-real-compact. Proof. Let α : A → R be an algebra homomorphism. We first show that for any finite set F ⊂ A there exists a point x ∈ X with f(x) = α(f) for all f ∈ F . For f ∈ A let Z(f) := {x ∈ X : f(x) = α(f)}. Then Z(f) = Z(f − α(f)1), since α(f − α(f)1) = 0. Hence we may assume that all f ∈ F are even contained in kerα = {f : α(f) = 0}. Then ⋂ f∈F Z(f) = Z( ∑ f∈F f ). The sets Z(f) are not empty, since otherwise f ∈ kerα and f(x) 6= 0 for all x, so 1 f ∈ A and hence 1 = f 1 f ∈ kerα, a contradiction to α(1) = 1. Now the lemma is valid, whether the condition “finitely generated” is meant in the sense of an ordinary algebra or even as an algebra with the additional assumption on non-vanishing functions, since then any f ∈ A can be written as a rational function in the elements of F . Thus α applied to such a rational function is just the rational function in the corresponding elements of α(F) = F(x), and is thus the value of the rational function at x. 2. Corollary. Any algebra-homomorphism α : A → R is monotone. Proof. Let f1 ≤ f2. By 1 there exists an x ∈ X such that α(fi) = fi(x) for i = 1, 2. Thus α(f1) = f1(x) ≤ f2(x) = α(f2). 3. Corollary. Any algebra-homomorphism α : A → R is bounded, for every convenient algebra structure on A. By a convenient algebra structure we mean a convenient vector space structure for which the multiplication A×A → A a bilinear bornological mapping. A convenient vector space is a separated locally convex vector space which is Mackey complete, see [FK]. Proof. Suppose that fn is a bounded sequence, but |α(fn)| is unbounded. Replacing fn by f 2 n we may assume that fn ≥ 0 and hence also α(fn) ≥ 0. Choosing a subsequence we may even assume that α(fn) ≥ 2 . Now consider ∑ n 1 2 fn. This series converges in the sense of Mackey, and since the bornology on A is complete the limit is an element f ∈ A. MORE SMOOTHLY REAL COMPACT SPACES 3