@ Appl. Gen. Topol. 22, no. 1 (2021), 193-198doi:10.4995/agt.2021.14585
© AGT, UPV, 2021

Duality of locally quasi-convex convergence

groups

Pranav Sharma

School of Basic Sciences, IIMT University, Meerut, Uttar Pradesh 250001. India. (pranav15851@gmail.com)

Communicated by F. Mynard

Abstract

In the realm of the convergence spaces, the generalisation of topological

groups is the convergence groups, and the corresponding extension of

the Pontryagin duality is the continuous duality. We prove that local

quasi-convexity is a necessary condition for a convergence group to be c-

reflexive. Further, we prove that every character group of a convergence

group is locally quasi-convex.

2010 MSC: 43A40; 54A20; 54H11.

Keywords: continuous duality; convergence groups; local quasi-convexity;

Pontryagin duality.

1. Introduction and Preliminaries

The character group Ĝ of an abelian topological group G is the group of all
continuous characters equipped with the compact-open topology. Pontryagin
duality theorem states that for any locally compact abelian (LCA) topological

group, the canonical map αG : G →
ˆ̂
G (from the group to the double dual

group) is a topological isomorphism. Mart́ın-Peinador [10] proves that if G is

reflexive and the evaluation map e : Ĝ × G → T defined as e(χ, x) = χ(x) is
continuous, then the group G is locally compact. This result explains the role
of local compactness in the Pontryagin duality theory. This result is further
generalized in [11, Theorem 1.1] where the same statement with the condition
“reflexive” replaced by the “quasi-convex compactness property” is proved.

Received 06 November 2020 – Accepted 07 January 2021

http://dx.doi.org/10.4995/agt.2021.14585


P. Sharma

In the last seventy years, extending the Pontryagin duality theory beyond
local compactness has gained the attention of several researchers [8, 13]. One
of these approaches is the continuous duality theory [5]. The compact-open
topology on the continuous character group is replaced with the continuous
convergence structure, and as a consequence of this, the evaluation map is
always continuous. A comparison of this approach with the Pontryagin duality
is presented in [7].

On the other hand, there are several situations in analysis (like convergence
in measure) where non-topological convergence originates, and the correspond-
ing generalisation of a topological space is the convergence space. In the realm
of the convergence spaces, the generalisation of topological groups is the con-
vergence groups. For details regarding the continuous duality for convergence
groups refer [4, 6]. Further, the notion of locally quasi-convex convergence
groups is introduced in [12]. Here, we prove that local quasi-convexity is a nec-
essary condition for a convergence group to be c-reflexive, and then we prove
that every character group of a convergence group is locally quasi-convex.

Before proceeding further, we present certain terms and notations required
for the rest of the article.

A filter F is a non-empty family of non-empty subsets of a set X which is
closed under supersets and finite intersection. We denote the set of all filters
on a set X by FX. Further, a subset H of a filter F is called basis of the filter
if each set in F contains a set in H. If F and G are filters on a set X then, F
is called coarser than G if F ⊂ G. Let λ be an arbitrary relation between X
and FX. The relation is called convergence (and the pair (X, λ) a convergence
space) on X if for F1, F2 in FX and x in X the following conditions hold:
(i) Centred: x↑ ∈ λ(x),
(ii) Isotone: If F1 ∈ λ(x) and F1 ≤ F2, then F2 ∈ λ(x) , and
(iii) Finitely deep: If F1, F2 ∈ λ(x), then F1 ∩ F2 ∈ λ(x).
This relation is also denoted by F −→

λ
x.

A convergence space X is Hausdorff if every filter on X converges to at most
one point. Further, a map f : (X, λ1) → (Y, λ2) between two convergence
spaces is said to be continuous if (F −→

λ1
x ⇒ f(F) −→

λ2
f(x)).

For any convergence spaces (X, λ1) and (Y, λ2) let C(X, Y ) denote the set of
all continuous functions from X to Y . The evaluation mapping e : C(X, Y ) ×
X → Y is defined as

e(f, x) = f(x) ∀ f ∈ C(X, Y ) and x ∈ X.

The continuous convergence structure on C(X, Y ) is defined as: a filter G
converge to f in C(X, Y ) iff e(G × F) converge to f(x) in Y , whenever F
converge to x in X. The space C(X, Y ) equipped with continuous convergence
structure is denoted by Cc(X, Y ).

A convergence group is a group with a compatible (the group operations
are continuous in the sense of convergence) convergence structure. The class

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 194



Duality of locally quasi-convex convergence groups

of convergence groups contains the class of topological groups. Some other
examples include the underlying groups of the convergence vector spaces.

For details about convergence spaces and convergence groups, we refer the
reader to [3, 9].

For a convergence abelian group (G, λ) denote by ΓG the set of all con-
tinuous homomorphisms of G into the circle group T. The set of all contin-
uous homomorphisms with the structure of continuous convergence is defined
as convergence dual of G and is denoted as (ΓG, λc). The evaluation map
κ : G → ΓΓG defined as κ(g)(χ) = χ(g) ∀ g ∈ G, χ ∈ ΓG is a continuous
group homomorphism. The convergence group is c-reflexive if this evaluation
map κ is a bicontinuous isomorphism.

As proved in [3, Example 8.5.14], there exists a locally compact non-reflexive
convergence group. Therefore, the analog of Pontryagin duality theorem, valid
in the context of abelian topological groups, cannot be directly extended to
abelian convergence groups. Motivated from the notion of local quasi-convexity
[2] in topological groups, the notion of local quasi-convexity for convergence
groups is defined in [12].

For a convergence abelian group G we define the polar and the inverse polar
of subsets of G and of ΓG as follows:

Definition 1.1 (Polar and inverse polar). For any subset H of G and L of ΓG
the polar and the inverse polar of H and L respectively are subsets defined as:

H✄ = {χ ∈ ΓG : χ(H) ⊂ T+}; L
✁ = {g ∈ G : χ(g) ⊂ T+, ∀ χ ∈ L},

here T+ = {z ∈ T : Re (z) ≥ 0}.

The local quasi-convexity for convergence groups is defined as:

Definition 1.2 (Quasi-convex set [3]). A subset A of a convergence abelian
group G is quasi-convex if for each point g in G\A, there is a character χ in
the polar set of A such that Reχ(g) < 0, that is A✄✁ = A.

Proposition 1.3. Let G be a convergence group, H ⊂ G and L ⊂ ΓG. The
polar H⊲ and the inverse polar L⊳ are quasi convex subsets of ΓcG and G
respectively.

Definition 1.4 (Locally quasi-convex convergence group [12]). A convergence
group G is locally quasi-convex if for each filter F −→

G
0 , there exists another

filter G coarser than F such that G −→
G

0 and G has a filter base composed of

quasi-convex sets.

2. Main results

Proposition 2.1. For a filter U on a convergence group G and for all U, V ∈ U
we have (U ∩ V )⊲⊲ ⊆ U⊲⊲ ∩ V ⊲⊲.

Proof. The proof is trivial. �

Proposition 2.2. {U⊲⊲ : U ∈ U} is a basis of a filter in ΓcΓcG.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 195



P. Sharma

Proof. The proof is similar to [3, Theorem 8.4.3]. �

We denote the filter generated by {U⊲⊲ : U ∈ U} as U⊲⊲.
Similarly, {U⊲⊳ : U ∈ U} is a basis of a filter in G which we denote by U⊲⊳.

Lemma 2.3. For a convergence abelian group G, the following statements hold:

(1) If a filter Φ −−−→
ΓcG

0, then A✄ ∈ Φ for every finite subset A of G.

(2) For every filter F −→
G

0, there is B ∈ F such that B✄ ∈ Φ.

Proof. The proof follows from [3, Proposition 8.1.8]. �

Theorem 2.4. For a convergence abelian group G, if U −→
G

0, then U✄✄ −−−−→
ΓcΓcG

0.

Proof. Applying Lemma 2.3 to ΓcG we need to prove that

a. A⊲ ∈ U⊲⊲ for each finite set A ⊆ ΓcG, and

b. for each filter Φ which converges to 0 in ΓcG, there is some P ∈ Φ such that
P ⊲ ∈ U⊲⊲.

a) Let A = {χ1, ..., χn}, then χi(U) → 0 for all i = 1, . . . , n. Further, there
are Ui ∈ U such that χi(Ui) ⊆ T+. If U = U1 ∩ ... ∩ Un, then χi(U) ⊆ T+
for all i which implies A ⊆ U⊲. Finally we have, U⊲⊲ ⊆ A⊲ and hence,
A⊲ ∈ U⊲⊲.

b) If Φ → 0 in ΓcG, then Φ(U) → 0 in T. Further, there are P ∈ Φ and U ∈ U
such that P(U) ⊆ T+. This gives P ⊆ U

⊲ which implies U⊲⊲ ⊆ P ⊲ and
hence, P ⊲ ∈ U⊲⊲.

�

Theorem 2.5. If a convergence group G is c-reflexive then it must be locally
quasi-convex.

Proof. To prove this result it is sufficient to prove that if a convergence group
is embedded then it must be locally quasi-convex.
In a convergence abelian group G let,

κG : G → ΓcΓcG

be an embedding. If U → 0 in G then by Theorem 2.4, we have, U⊲⊲ → 0 in
ΓcΓcG and so κ

−1(U⊲⊲) → 0 in G. Now we have

κ−1(U⊲⊲) = U⊲⊳ ⊆ U.

In view of Proposition 1.3 the proof follows. �

Next we prove that the continuous character group of a convergence group
is locally quasi-convex.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 196



Duality of locally quasi-convex convergence groups

Lemma 2.6. Let X be a convergence space and A is a subset of X. Further-
more, let G be a convergence group and M ⊆ G a quasi-convex set. Then

P(A, M) = {g ∈ Cc(X, G) : g(A) ⊂ M}

is quasi-convex in the group Cc(X, G).

Proof. Take any g0 6∈ P(A, M). Then there is a point x0 ∈ A such that
g0(x0) 6∈ M. Since M is quasi-convex there is some χ ∈ Γ(G) such that
χ(M) ⊂ T+ while χ(g0(x0)) 6∈ T+. Define φ : Cc(X, G) → T by

φ(g) = χ(g(x0)).

Then φ is a continuous character, φ(g0) = χ(g0(x0)) 6∈ T+ while φ(g) =
χ(g(x0)) ∈ χ(M) ⊆ M for all g ∈ P(A, M). �

The next lemma is an extension of [1, Proposition 6.2, (ii)] to the realm of
convergence groups.

Lemma 2.7. Let G be a convergence group and B be a family of quasi-convex
subsets of G. Then

B0 =
⋂

{B : B ∈ B}

is a quasi-convex subset of G.

Proof. A direct proof is easy, so omitted. �

Theorem 2.8. Let X be a convergence space and G be a locally quasi-convex
topological group. Then Cc(X, G) is locally quasi-convex.

Proof. Choose a zero neighbourhood basis B in G consisting of quasi-convex
sets and take a filter G on Cc(X, G) which converges to 0. Then G(Φ) → 0 ∈ G
for each x ∈ X, and each filter Φ on X which converges to x. So for each B ∈ B
there are Gx,Φ,B ∈ G and Fx,Φ,B ∈ Φ such that

Gx,Φ,B(Fx,Φ,B) ⊆ B.

Set

Hx,Φ,B = P(Fx,Φ,B, B)

then {Hx,Φ,B : Φ → x, B ∈ B} is the subbasis of a filter H which converges to 0
in Cc(X, G). By the Lemma 2.6 and Lemma 2.7, H has a base of quasi-convex
sets and H ⊆ G since Gx,Φ,B ⊆ Hx,Φ,B for all x, Φ, B. �

Corollary 2.9. For each convergence group G, ΓcG is locally quasi-convex.

Proof. Since T and Cc(G, T) are locally quasi-convex so is Γc(G) as a subgroup.
�

Acknowledgements. We thank Prof. H.-P. Butzmann and the anonymous

reviewers for their many insightful comments and suggestions.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 197



P. Sharma

References

[1] L. Außenhofer, Contributions to the Duality Theory of Abelian Topological Groups and
to the Theory of Nuclear Groups, Dissertationes mathematicae. Institute of Mathemat-
ics, Polish Academy of Sciences, 1999.

[2] W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in
Mathematics, Springer Berlin Heidelberg, 1991.

[3] R. Beattie and H.-P. Butzmann, Convergence Structures and Applications to Functional
Analysis, Bücher, Springer Netherlands, 2013.

[4] M. Bruguera, Topological groups and convergence groups: Study of the Pontryagin
duality, Thesis, 1999.

[5] H.-P. Butzmann, Über diec-Reflexivität von Cc(X), Comment. Math. Helv. 47, no. 1
(1972), 92–101.

[6] H.-P. Butzmann, Duality theory for convergence groups, Topology Appl. 111, no. 1
(2000), 95–104.

[7] M. J. Chasco and E. Mart́ın-Peinador, Binz-Butzmann duality versus Pontryagin duality,
Arch. Math. (Basel) 63, no. 3 (1994), 264–270.

[8] M. J. Chasco, D. Dikranjan and E. Mart́ın-Peinador, A survey on reflexivity of abelian
topological groups, Topology Appl. 159, no. 9 (2012), 2290–2309.

[9] S. Dolecki and F. Mynard, Convergence Foundations of Topology, World Scientific Pub-
lishing Company, 2016.

[10] E. Mart́ın-Peinador, A reflexive admissible topological group must be locally compact,
Proc. Amer. Math. Soc. 123, no. 11 (1995), 3563–3566.

[11] E. Mart́ın-Peinador and V. Tarieladze, A property of Dunford-Pettis type in topological
groups, Proc. Amer. Math. Soc. 132, no. 6 (2004), 1827–1837.

[12] P. Sharma, Locally quasi-convex convergence groups, Topology Appl. 285 (2020),
107384.

[13] P. Sharma and S. Mishra, Duality in topological and convergence groups, Top. Proc., to

appear.

© AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 198