Tensor products of topological abelian groups and Pontryagin duality
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911769106055168 |
|---|---|
| author | Ferrer, María V. Hernández-Arzusa, Julio Hernández, Salvador |
| author_facet | Ferrer, María V. Hernández-Arzusa, Julio Hernández, Salvador |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_01223 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Tensor products of topological abelian groups and Pontryagin duality Ferrer, María V. Hernández-Arzusa, Julio Hernández, Salvador Functional Analysis General Topology Group Theory 22A05 (22A25 22E99 20C15 20K20 54H11 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. |
| title | Tensor products of topological abelian groups and Pontryagin duality |
| topic | Functional Analysis General Topology Group Theory 22A05 (22A25 22E99 20C15 20K20 54H11 |
| url | https://arxiv.org/abs/2309.01223 |