Tensor products of topological abelian groups and Pontryagin duality

TL;DR

This paper investigates the duality and reflexivity properties of certain topological abelian groups, including groups of homomorphisms and continuous functions, revealing their dual groups as tensor products and identifying conditions for reflexivity.

Contribution

It explicitly determines the dual group of a specific non-reflexive prodiscrete group and explores reflexivity conditions for groups of continuous functions and free abelian groups on 0-dimensional spaces.

Findings

The dual group of G is topologically isomorphic to the tensor product of ZZ^\u007f and TT.

Identifies conditions under which groups of continuous functions are Pontryagin reflexive.

Provides explicit duality descriptions for free abelian groups on 0-dimensional spaces.

Abstract

Let $G$ be the group of all $\ZZ$-valued homomorphisms of the Baer-Specker group $\ZZ^\NN$. The group $G$ is algebraically isomorphic to $\ZZ^{(\NN)}$, the infinite direct sum of the group of integers, and equipped with the topology of pointwise convergence on $\ZZ^\NN$, becomes a non reflexive prodiscrete group. It was an open question to find its dual group $\hat{G}$. Here, we answer this question by proving that $\hat{G}$ is topologically isomorphic to $\ZZ^\NN\otimes_\mathcal{Q}\TT$, the (locally quasi-convex) tensor product of $\ZZ^\NN$ and $\TT$. Furthermore, we investigate the reflexivity properties of the groups of $C_p(X,\ZZ)$, the group of all $\ZZ$-valued continuous functions on $X$ equipped with the pointwise convergence topology, and $A_p(X)$, the free abelian group on a $0$-dimensional space $X$ equipped with the topology $t_p(C(X,\ZZ))$ of pointwise convergence topology on $C(X,\ZZ)$. In particular, we prove that $\hat{A_p(X)}\simeq C_p(X,\ZZ)\otimes_\mathcal{Q}\TT$ and we establish the existence of $0$-dimensional spaces $X$ such that $C_p(X,\ZZ)$ is Pontryagin reflexive.