#
Krein’s Theorem in the Context of Topological Abelian Groups^{ †}

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

#### Preliminaries

## 2. Generalities on the Quasi-Convex Compactness Property

**Definition**

**1.**

**Proposition**

**1.**

- (a)
- Every semi-reflexive locally quasi-convex group has the qcp.
- (b)
- Every complete locally quasi-convex group has the qcp.
- (c)
- A locally quasi-convex group with the qcp can fail to be semi-reflexive. Actually there exists a complete, metrizable, locally quasi-convex group which is not semi-reflexive.
- (d)
- A locally quasi-convex group with the qcp can fail to be complete.
- (e)
- A metrizable, locally quasi-convex group with the qcp is necessarily complete.
- (f)
- If X is a topological abelian group such that ${\alpha}_{X}:X\to {\left({X}_{c}^{\wedge}\right)}_{c}^{\wedge}$ is continuous, then ${X}_{c}^{\wedge}$ has the qcp.
- (g)
- If σ and τ are compatible locally quasi-convex group topologies on an abelian group X where $\sigma \le \tau ,$ and $(X,\sigma )$ has the qcp, then $(X,\tau )$ has the qcp too.

**Proof.**

- (c)
- Such an example can be found in ([3], Corollary 11.15). Note that this group has the qcp by (a).
- (d)
- Let G be any locally compact, noncompact abelian group. Put $X=(G,\sigma (G,{G}^{\wedge})).$ By Glicksberg’s Theorem, ${X}_{c}^{\wedge}={G}_{c}^{\wedge}.$ This implies, on the one hand, that ${\left({X}_{c}^{\wedge}\right)}^{\wedge}=G,$ that is, X is semi-reflexive and by (a) has the qcp. On the other hand, X is not complete since otherwise it would be compact and in particular ${\left({X}_{c}^{\wedge}\right)}_{c}^{\wedge}={\left({G}_{c}^{\wedge}\right)}_{c}^{\wedge}\cong G$ would be compact as well, a contradiction.
- (e)

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Problem**

**1.**

**Lemma**

**2.**

**Proof.**

**Proposition**

**2.**

- (a)
- X has the qcp.
- (b)
- ${\mathcal{T}}_{c}\le {\mathcal{T}}_{\sigma qc}$.

**Proof.**

## 3. The qcp on Subgroups

**Proposition**

**3.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**2.**

**Corollary**

**3.**

**Corollary**

**4.**

**Lemma**

**3.**

**Lemma**

**4.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Example**

**1.**

**Theorem**

**4.**

- (i)
- ${e}_{H}:{H}_{c}^{\wedge}\times H\to \mathbb{T}$ is continuous.
- (ii)
- H is locally precompact and determines its completion.

**Proof.**

**Corollary**

**5.**

**Remark**

**1.**

**Problem 2.**

- (i)
- If a topological group X contains a dense subgroup which is a k-space and determines X, must X be a k-space?
- (ii)
- If a topological group X contains a dense subgroup H which is a k-space, does H determine X?

**Corollary**

**6.**

**Proof.**

**Remark**

**2.**

**Example**

**2.**

**Problem**

**3.**

## 4. The qcp in Topological Vector Spaces

**Proposition**

**6.**

- (a)
- For every compact subset K of E, the set ${\mathrm{qc}}_{E}\left(K\right)$ is compact (i.e., E has the qcp).
- (a′)
- For every compact subset K of E, the set ${\mathrm{qc}}_{E}\left(K\right)$ is $\sigma (E,{E}^{\wedge})\u2013$compact.
- (b)
- For every compact subset K of E, the set ${K}^{\circ \circ}$ is compact.
- (b′)
- For every compact subset K of E, the set ${K}^{\circ \circ}$ is $\omega (E,{E}^{\ast})\u2013$compact.
- (c)
- For every compact subset K of E, the set ${\mathrm{acc}}_{E}\left(K\right)$ is compact (i.e., E has the ccp).
- (d)
- The natural mapping ${\alpha}_{E}:E\to {\left({E}_{c}^{\wedge}\right)}^{\wedge}$ defined by ${\alpha}_{E}\left(x\right)\left(\chi \right)=\chi \left(x\right)$ is onto (i.e., E is semi-reflexive as a topological abelian group).
- (e)
- The natural mapping ${\gamma}_{E}:E\to {\left({E}_{c}^{\ast}\right)}^{\ast}$ defined by ${\gamma}_{E}\left(x\right)\left(f\right)=f\left(x\right)$ is onto.

**Proof.**

**Proposition**

**7.**

- (a)
- E has the qcp.
- (b)
- E has the ccp.
- (c)
- E is locally convex and complete.

**Proof.**

**Example**

**3.**

**Proposition**

**8.**

- (a)
- The group $(E,\sigma (E,{E}^{\wedge}))$ has the qcp.
- (b)
- The space $(E,\omega (E,{E}^{\ast}))$ has the ccp.
- (c)
- $(E,\sigma (E,{E}^{\wedge}))$ is a semi-reflexive group.

**Proof.**

- (a′)
- $(E,\omega (E,{E}^{\ast}))$ has the qcp and on the other hand that

- (b′)
- $(E,\omega (E,{E}^{\ast}))$ is semi-reflexive as a topological abelian group.

## 5. The Krein Property for Topological Abelian Groups

**Theorem**

**5.**

**Definition**

**2.**

**Proposition**

**9.**

- (a)
- X has the Krein property.
- (b)
- The topologies ${\mathcal{T}}_{\sigma c}$ and ${\mathcal{T}}_{\sigma qc}$ coincide on ${X}^{\wedge}.$

**Proposition**

**10.**

**Proposition**

**11.**

**not**remain true for complete locally quasi-convex groups, we present a family of counterexamples considered in [30] with a different purpose.

**Notation**

**1.**

**Theorem**

**6.**

**Proof.**

## 6. The Krein and the Glicksberg Properties in the Context of Duality

**Proposition**

**12.**

- (a)
- X has the Glicksberg property.
- (b)
- ${\mathcal{T}}_{c}={\mathcal{T}}_{\sigma c}={\mathcal{T}}_{\sigma qc}$.

**Proof.**

**Proposition**

**13.**

- (a)
- ${X}_{c}^{\wedge}$ is g-barrelled.
- (b)
- X has the Glicksberg property.

**Proof.**

**Corollary**

**7.**

- (a)
- ${X}_{c}^{\wedge}$ is g-barrelled.
- (b)
- X has the Glicksberg property.

**Remark**

**3.**

**Corollary**

**8.**

- (i)
- ${X}_{c}^{\wedge}$ is g-barrelled.
- (ii)
- X has the Glicksberg property.
- (iii)
- The topologies ${\mathcal{T}}_{c}$ and ${\mathcal{T}}_{\sigma qc}$ coincide on ${X}^{\wedge}.$

**Example**

**4.**

- (i)
- Banach spaces provide examples of reflexive topological groups with the Krein property. Just take into account that a Banach space is a reflexive topological group ([6]), and Theorem 5 and Proposition 11 of the present paper.
- (ii)
- A reflexive group $(G,\tau )$ with the Krein property, such that ${G}_{c}^{\wedge}$ is not g-barrelled: Let G be an infinite dimensional, reflexive Banach space (in the ordinary sense of reflexivity for Banach spaces). It does not have the Glicksberg property: in fact, the unit ball B is $\omega (G,{G}^{\ast})$-compact and by [24] also $\sigma (G,{G}^{\wedge})$-compact. Clearly B is not compact in the norm topology of G. Thus, Corollary 8 applies to obtain that ${G}_{c}^{\wedge}$ is not g-barrelled.
- (iii)
- A non reflexive group with Krein and Glicksberg properties such that ${G}_{c}^{\wedge}$ is g-barrelled: Let $G:=(E,\omega (E,{E}^{\ast}))$ where E is an infinite dimensional Banach space and $\omega (E,{E}^{\ast})$ is its weak topology. The group G is locally quasi-convex nonreflexive (${\alpha}_{G}$ is not continuous) and by (i) it has the Krein property. Since the $\omega (E,{E}^{\ast})$-compact subsets of E coincide with the $\sigma (E,{E}^{\wedge})$-compact subsets ([24], Lemma 1.2), G has the Glicksberg property. By Proposition 12, the compact-open topology on ${G}^{\wedge}$ coincides with ${\mathcal{T}}_{\sigma qc}$.By Proposition 8, G is semi-reflexive and Proposition 13 proves that ${G}_{c}^{\wedge}$ is g-barrelled. Observe also that G itself is not g-barrelled.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Grothendieck, A. Topological Vector Spaces; Gordon and Breach: New York, NY, USA; London, UK; Paris, France, 1973. [Google Scholar]
- Ostling, E.G.; Wilansky, A. Locally convex topologies and the convex compactness property. Proc. Camb. Philos. Soc.
**1974**, 75, 45–50. [Google Scholar] [CrossRef] - Aussenhofer, L. Contributions to the Duality Theory of Abelian Topological Groups and to the Theory of Nuclear Groups; Dissertationes Mathematicae; PWN: Warszawa, Poland, 1999; Volume 384. [Google Scholar]
- Banaszczyk, W.; Chasco, M.J.; Martín Peinador, E. Open subgroups and Pontryagin duality. Math. Z.
**1994**, 215, 195–204. [Google Scholar] [CrossRef] - Chasco, M.J. Pontryagin duality for metrizable groups. Arch. Math.
**1998**, 70, 22–28. [Google Scholar] [CrossRef] - Smith, M.F. The Pontrjagin Duality Theorem in Linear Spaces. Ann. Math.
**1952**, 56, 248–253. [Google Scholar] [CrossRef] - Banaszczyk, W. Additive subgroups of topological vector spaces. In Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 1991; Volume 1466. [Google Scholar]
- Robertson, A.P.; Robertson, W. Topological Vector Spaces, 2nd ed.; Cambridge University Press: London, UK; New York, NY, USA, 1973. [Google Scholar]
- Hofmann, K.H.; Morris, S.A. The Structure of Compact Groups, 4th ed.; Studies in Mathematics; De Gruyter: Berlin, Germany; Boston, MA, USA, 2020; Volume 25. [Google Scholar]
- Bruguera, M. Grupos Topológicos y Grupos de Convergencia: Estudio de la Dualidad de Pontryagin. Ph.D. Thesis, University of Barcelona, Barcelona, Spain, 1999. [Google Scholar]
- Bruguera, M.; Martín-Peinador, E. Banach-Dieudonné theorem revisited. J. Aust. Math. Soc.
**2003**, 75, 1–15. [Google Scholar] [CrossRef][Green Version] - Hernández, S. Pontryagin duality for topological abelian groups. Math. Z.
**2001**, 238, 493–503. [Google Scholar] [CrossRef] - Lukács, G. Notes on duality theories of abelian groups. arXiv
**2006**, arXiv:math/0605149. [Google Scholar] - Araki, T. A characterization of non-local convexity in some class of topological vector spaces. Math. Japon.
**1995**, 41, 573–577. [Google Scholar] - Morris, S.A. Pontryagin Duality and the Structure of Locally Compact Abelian Groups; London Mathematical Society Lecture Notes Series; Cambridge University Press: Cambridge, UK; London, UK; New York, NY, USA; Melbourne, Australia, 1977; Volume 29. [Google Scholar]
- Martín-Peinador, E.; Tarieladze, V. A property of Dunford-Pettis type in topological groups. Proc. Amer. Math. Soc.
**2003**, 132, 1827–1837. [Google Scholar] [CrossRef] - Martín-Peinador, E. A reflexive admissible topological group must be locally compact. Proc. Am. Math. Soc.
**1995**, 123, 3563–3566. [Google Scholar] [CrossRef][Green Version] - Comfort, W.W.; Raczkowki, S.U.; Trigos-Arrieta, F.J. The dual group of a dense subgroup. Czechoslovak Math. J.
**2004**, 54, 509–533. [Google Scholar] [CrossRef][Green Version] - Turnwald, G. On the continuity of the evaluation mapping associated with a group and its character group. Proc. Amer. Math. Soc.
**1998**, 126, 3413–3415. [Google Scholar] [CrossRef] - Noble, N. The continuity of functions on Cartesian products. Trans. Amer. Math. Soc.
**1970**, 149, 187–198. [Google Scholar] [CrossRef] - Engelking, R. General Topology; Sigma Series in Pure Mathematics; Heldermann Verlag: Berlin, Germany, 1989; Volume 6. [Google Scholar]
- Kelley, J.L. General Topology; University Series in Higher Mathematics; D. Van Nostrand: Toronto, ON, Canada; New York, NY, USA; London, UK, 1955. [Google Scholar]
- Bruguera, M.; Tkachenko, M. Pontryagin duality in the class of precompact Abelian groups and the Baire property. J. Pure Appl. Algebra
**2012**, 216, 2636–2647. [Google Scholar] [CrossRef] - Remus, D.; Trigos-Arrieta, F.J. Abelian groups which satisfy Pontryagin duality need not respect compactness. Proc. Amer. Math. Soc.
**1993**, 117, 1195–1200. [Google Scholar] [CrossRef] - Chasco, M.J.; Martín-Peinador, E.; Tarieladze, V. On Mackey topology for groups. Studia Math.
**1999**, 132, 257–284. [Google Scholar] [CrossRef] - Wilansky, A. Modern methods in Topological Vector Spaces; McGraw-Hill International Book Co.: New York, NY, USA, 1978. [Google Scholar]
- Mazur, S.; Orlicz, W. Sur les espaces métriques linéaires (I). Stud. Math.
**1948**, 10, 184–208. [Google Scholar] [CrossRef][Green Version] - Kalton, N.J.; Peck, N.T.; Roberts, J.W. An F-Space Sampler; London Mathematical Society Lecture Note Series; Cambridge University Press: Cambridge, UK; New York, NY, USA; Melbourne, Australia, 1984; Volume 89. [Google Scholar]
- Schaefer, H.H. Topological Vector Spaces; Graduate Texts in Mathematics; Springer: New York, NY, USA; Heidelberg/Berlin, Germany, 1971; Volume 3. [Google Scholar]
- Dikranjan, D.; Martín-Peinador, E.; Tarieladze, V. Group valued null sequences and metrizable non-Mackey groups. Forum Math.
**2014**, 26, 723–757. [Google Scholar] [CrossRef][Green Version] - Venkataraman, R. Compactness in abelian topological groups. Pac. J. Math.
**1975**, 57, 591–595. [Google Scholar] [CrossRef] - Gabriyelyan, S. Maximally almost periodic groups and respecting properties. In Descriptive Topology and Functional Analysis II; Ferrando, J.C., Ed.; Springer Proceedings in Mathematics & Statistics; Springer: Cham, Switzerland, 2019; Volume 286, pp. 103–106. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Borsich, T.; Domínguez, X.; Martín-Peinador, E.
Krein’s Theorem in the Context of Topological Abelian Groups. *Axioms* **2022**, *11*, 224.
https://doi.org/10.3390/axioms11050224

**AMA Style**

Borsich T, Domínguez X, Martín-Peinador E.
Krein’s Theorem in the Context of Topological Abelian Groups. *Axioms*. 2022; 11(5):224.
https://doi.org/10.3390/axioms11050224

**Chicago/Turabian Style**

Borsich, Tayomara, Xabier Domínguez, and Elena Martín-Peinador.
2022. "Krein’s Theorem in the Context of Topological Abelian Groups" *Axioms* 11, no. 5: 224.
https://doi.org/10.3390/axioms11050224