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

Abstract

From the introduction: "For a completely regular space X , C ∗ (X) denotes the algebra of all bounded real-valued continuous functions over X . We consider the topology of uniform convergence over C ∗ (X) .

"In this paper we carry out a systematic study of uniform approximation for algebras and lattices of C ∗ (X) . If F is an algebra or lattice (vector lattice, affine lattice,… ) we characterize its uniform closure and we give necessary and sufficient conditions for uniform density in C ∗ (X) . For our purposes we identify the rings C ∗ (X) and C(βX) (βX is the Stone-Čech compactification of X ). In this way, we generalize the classical results in the compact case and also obtain the generalizations by Hewitt and Blasco