Stone-Weierstrass theorems for group-valued functions

Constructive groups were introduced by Sternfeld in [6] as a class of metrizable groups G for which a suitable version of the Stone-Weierstrass theorem on the group of G-valued functions C(X, G) remains valid. As a way of exploring the existence of such Stone-Weierstrass-type theorems in this context we address the question raised in [6] as to which groups are constructive and prove that a locally compact group with more than two elements is constructive if and only if it is either totally disconnected or homeomorphic to some vector group R-n. It may therefore be concluded that the Stone-Weierstrass theorem can be extended to some noncommutative Lie groups - exactly to those not containing any nontrivial compact subgroup.