Tensor products of topological abelian groups and Pontryagin duality

Abstract

Let G be the group of all Z-valued homomorphisms of the Baer-Specker group ZN. The group G is algebraically isomorphic to Z(N), the infinite direct sum of the group of integers, and equipped with the topology of pointwise convergence on ZN, becomes a non reflexive prodiscrete group. It was an open question to find its dual group Gˆ. Here, we answer this question by proving that Gˆ is topologically isomorphic to ZN⊗QT, the (locally quasi-convex) tensor product of ZN and T. Furthermore, we investigate the reflexivity properties of the groups Cp(X,Z), the group of all Z-valued continuous functions on X equipped with the pointwise convergence topology, and Ap(X), the free abelian group on a 0-dimensional space X equipped with the topology tp(C(X,Z)) of pointwise convergence topology on C(X,Z). In particular, we prove that Ap(X)ˆ≃Cp(X,Z)⊗QT and we establish the existence of 0-dimensional spaces X such that Cp(X,Z) is Pontryagin reflexive.