Pontryagin duality for spaces of continuous functions

A topological abelian group G is P-reflexive if the natural homomorphism of G to its Pontryagin bidual group is a topological isomorphism. Let C-p(X) be the space of continuous functions with the topology of pointwise converg ence. We investigate for what spaces X the group C-p(X) is P-reflective. We show that: (1) if C-p(X) is P-reflexive, then X is a P-space; (2) there ex ists a non-discrete space X such that C-p(X) is P-reflexive; (3) there exis ts a P-space X such that C-p(X) is not P-reflexive; (4) there exists a simp le space X for which the question of whether C-p(X) is P-reflexive is undec idable in ZFC, (C) 2000 Academic Press.