Stone-weierstrass theorems for group-valued functions

Constructive groups were introduced by Sternfeld in 6] as a class of metrizable groupsG for which a suitable version of the Stone-Weierstrass theorem on the group ofG-valued functionsC(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. Research partially supported by Grant CTIDIB/2002/192 of theAgencia Valenciana de Ciencia y Tecnología, and Fundació Caixa-Castelló, grant P1 B2001-08.