Pontryagin Duality for Spaces of Continuous Functions

Abstract

A topological abelian group G is P-reflexive 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 group Cp(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 that Cp(X) is P-reflexive; (3) there exists a P-space X such that Cp(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.