Pontryagin Duality for Spaces of Continuous Functions

A topological abelian group G is P-reflexi¨e if the natural homomorphism of G to its Pontryagin bidual group is a topological isomorphism. Let Cp X. be the space of continuous functions with the topology of pointwise convergence. We investigate for what spaces X the groupCp X.is P-reflexive. We show that: 1.if Cp X.is P-reflexive, then X is a P-space; 2.there exists a non-discrete space X such thatCp X.is P-reflexive; 3.there exists a P-space X such thatCp X.is not P-reflexive; 4. there exists a simple space X for which the question of whether Cp X.is P-reflexive is undecidable in ZFC.