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Tensor products of topological abelian groups and Pontryagin duality

María V. Ferrer, Julio Hernández-Arzusa, Salvador Hernández

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.

Tensor products of topological abelian groups and Pontryagin duality

Abstract

Let be the group of all -valued homomorphisms of the Baer-Specker group . The group is algebraically isomorphic to , the infinite direct sum of the group of integers, and equipped with the topology of pointwise convergence on , becomes a non reflexive prodiscrete group. It was an open question to find its dual group . Here, we answer this question by proving that is topologically isomorphic to , the (locally quasi-convex) tensor product of and . Furthermore, we investigate the reflexivity properties of the groups of , the group of all -valued continuous functions on equipped with the pointwise convergence topology, and , the free abelian group on a -dimensional space equipped with the topology of pointwise convergence topology on . In particular, we prove that and we establish the existence of -dimensional spaces such that is Pontryagin reflexive.
Paper Structure (8 sections, 32 theorems, 57 equations)

This paper contains 8 sections, 32 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Tensor products in the category of topological abelian groups
  3. Locally quasi-convex groups
  4. Main result
  5. Basic facts
  6. The dual group of ${\mathbb Z}^{(I)}$
  7. The dual group of $A_p(X)$
  8. Groups of integer-valued continuous functions

Key Result

Theorem A

Let $G$ denote the group ${\mathbb Z}^{({\mathbb N})}\simeq\mathrm{Hom}({\mathbb Z}^{\mathbb N},{\mathbb Z})$, equipped with the topology $t_p({\mathbb Z}^{\mathbb N})$ (inherited from ${\mathbb Z}^{{\mathbb Z}^{\mathbb N}}$). Then we have $\widehat{G}\cong {\mathbb Z}^{\mathbb N}\otimes_\mathcal{Q}

Theorems & Definitions (45)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Remark 2.7
  • ...and 35 more