On some generalizations of the Kakutani-Stone and Stone-Weierstrass theorems

Abstract

All spaces are completely regular and Hausdorff. For a space X, C(X) denotes the algebra of all bounded continuous real valued functions defined on X. In [Duke Math.J. 14, 419-427 (1947; Zbl 0029.30302)] E. Hewitt gave a uniform density theorem for algebras of C(X) containing all the constant functions. J. L. Blasco [Topology Appl. 15, 111-117 (1983; Zbl 0498.54021)] obtained a relationship between an extension problem of bounded continuous functions and a problem of uniform approximation. In this paper the authors study uniform approximation for algebras and lattices of C(X). If F is an algebra or lattice (vector lattice, affine lattice, etc.), its uniform closure is characterized, and necessary and sufficient conditions are given for uniform density in C(X). They obtain theorems such as the following: Theorem 5. Let F be a sublattice of C(X) and let f 2 C(X). Suppose that for every a < b and " > 0 there exists g 2 F such that |g(x)−a| < " if f(x) a and |g(x)−b| < "if f(x) b. Then f 2 F. Theorem 11. Let F be a subalgebra of C(X). Then F is uniformly dense in C(X) if and only if the following hold: (i) F separates every pair of disjoint zero-sets in X; and (ii) F contains a unity of C(X) (i.e., there is f 2 F with f " > 0).