Search

We study gaps in the lattice of topological group topologies on a given abelian group and compare the properties of the topologies that form a gap. The emphasis is placed on the study of predecessors of the topology of a noncompact LCA group. It is shown that every Hausdorff predecessor, σ, of the topology τ of an LCA group G is finer than the Bohr topology of the group G and that the two topologies, τ and σ, have the same closed subgroups. This implies that the Bohr topology, τ + , of a noncompact LCA group ( G , τ ) is not a predecessor of τ. Complementing these results we prove that all predecessors of the topology of a Hausdorff topological abelian group G are Hausdorff provided that G is torsion free. We also show that if a topological abelian group ( G , τ ) contains a complete subgroup N such that the quotient group G / N is compact, then the predecessors of τ are in a one-to-one correspondence with the predecessors of τ ↾ N on N. In particular, if N is discrete and G / N is compact, then the predecessors of τ are in a one-to-one correspondence with the maximal topological group topologies on N.

Financiado para publicación en acceso aberto: Universidade da Coruña/CISUG [Abstract] Let X be a topological abelian group and H a subgroup of X. We find conditions under which local quasi-convexity of both H and results in the same property for X. This is true for instance if H is precompact, or if X is metrizable and H is a dually embedded subgroup which is also either discrete or bounded torsion. We also give some general principles and point out some errors we have found in the existing literature on this problem. The authors are grateful to M. Jesús Chasco and Elena Martín-Peinador for their very useful ideas and suggestions. The first author acknowledges the financial support of the Spanish AEI and FEDER UE funds (grants MTM2013-42486-P and MTM2016-79422-P). The second author was partially supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) grant no. DI-18-1429 Shota Rustaveli National Science Foundation of Georgia (SRNSFG); DI-18-1429

In this paper, we give conditions under which a commutative topological semigroup can be embedded algebraically and topologically into a compact topological Abelian group. We prove that every feebly compact regular first countable cancellative commutative topological semigroup with open shifts is a topological group, as well as every connected locally compact Hausdorff cancellative commutative topological monoid with open shifts. Finally, we use these results to give sufficient conditions on a commutative topological semigroup that guarantee it to have countable cellularity.

We show the classical $\pi_1$-action on the $n$-th homotopy group can fail to be continuous for any $n$ when the homotopy groups are equipped with the natural quotient topology. In particular, we prove the action $\pi_1(X)\times\pi_n(X)\to\pi_n(X)$ fails to be continuous for a one-point union $X=A\vee \mathbb{H}_n$ where $A$ is an aspherical space such that $\pi_1(A)$ is a topological group and $\mathbb{H}_n$ is the $(n-1)$-connected, n-dimensional Hawaiian earring space $\mathbb{H}_n$ for which $\pi_n(\mathbb{H}_n)$ is a topological abelian group., Comment: 13 pages

It is a simple fact that a subgroup generated by a subset $A$ of an abelian group is the direct sum of the cyclic groups $\langle a\rangle$, $a\in A$ if and only if the set $A$ is independent. In [5] the concept of an $independent$ set in an abelian group was generalized to a $topologically$ $independent$ $set$ in a topological abelian group (these two notions coincide in discrete abelian groups). It was proved that a topological subgroup generated by a subset $A$ of an abelian topological group is the Tychonoff direct sum of the cyclic topological groups $\langle a\rangle$, $a\in A$ if and only if the set $A$ is topologically independent and absolutely Cauchy summable. Further, it was shown, that the assumption of absolute Cauchy summability of $A$ can not be removed in general in this result. In our paper we show that it can be removed in precompact groups. In other words, we prove that if $A$ is a subset of a {\em precompact} abelian group, then the topological subgroup generated by $A$ is the Tychonoff direct sum of the topological cyclic subgroups $\langle a\rangle$, $a\in A$ if and only if $A$ is topologically independent. We show that precompactness can not be replaced by local compactness in this result.

We say that a (para)topological group G is strongly submetrizable if it admits a coarser separable metrizable (para)topological group topology and is projectively strongly submetrizable if for each open neighborhood U of the identity in G, there is a closed invariant subgroup N contained in U such that the quotient (para)topological group G / N is strongly submetrizable. We show that a quotient group of a simply sm-factorizable ω-narrow topological abelian group can fail to be simply sm-factorizable. This answers a question posed by Arhangel'skii and the first listed author in 2018. If, however, the kernel of a quotient homomorphism is a bounded subgroup, then the homomorphism preserves simple sm-factorizability in the classes of topological and paratopological groups. We also prove that a regular (para)topological group G is simply sm-factorizable if and only if G is projectively strongly submetrizable and every continuous real-valued function on G is uniformly continuous on G ω , the P-modification of G. Making use of this fact we show that all weakly Lindelof projectively strongly submetrizable paratopological groups and all weakly Lindelof paratopological abelian groups are simply sm-factorizable. It is also established that every precompact paratopological group is simply sm-factorizable.

We show that if Y is a dense subspace of a Tychonoff space X, then w ( X ) ≤ n w ( Y ) N a g ( Y ) , where N a g ( Y ) is the Nagami number of Y. In particular, if Y is a Lindelof Σ-space, then w ( X ) ≤ n w ( Y ) ω ≤ n w ( X ) ω . Better upper bounds for the weight of topological groups are given. For example, if a topological group H contains a dense subgroup G such that G is a Lindelof Σ-space, then w ( H ) = w ( G ) ≤ ψ ( G ) ω . Further, if a Lindelof Σ-space X generates a dense subgroup of a topological group H, then w ( H ) ≤ 2 ψ ( X ) . Several facts about subspaces of Hausdorff separable spaces are established. It is well known that the weight of a separable Hausdorff space X can be as big as 2 2 c . We prove on the one hand that if a regular Lindelof Σ-space Y is a subspace of a separable Hausdorff space, then w ( Y ) ≤ 2 ω , and the same conclusion holds for a Lindelof P-space Y. On the other hand, we present an example of a countably compact topological Abelian group G which is homeomorphic to a subspace of a separable Hausdorff space and satisfies w ( G ) = 2 2 c , i.e. G has the maximal possible weight.

It is proved that, on any Abelian group of infinite cardinality m, there exist precisely 2 2 m nonequivalent bounded Hausdorff group topologies. Under the continuum hypothesis, the number of nonequivalent compact and locally compact Hausdorff group topologies on the group ( Z p ) N is determined.

A detailed description of a normalized internal bicategory in the category of groups is derived from the general description of internal bicategories in weakly Mal’tsev categories endowed with a V-Mal’tsev operation in the sense of Pedicchio. The example of bicategory of paths in a topological abelian group is presented.

Denote by \({{\mathbb {T}}}\)the torus,i.e., the topological group consisting of the complex numbers of modulus 1 under multiplication. Every topological Abelian group (G, t) has associated a weaker topological group topology, denoted by \(t^+\), defined as the weakest topology on G that makes the t-continuous homomorphisms (t-characters) \(\phi : G \rightarrow {{\mathbb {T}}}\) continuous. The topology \(t^+\) is called the Bohr topology on (G, t). Let \({{{\mathfrak {T}}}}\) denote the torsion subgroup of \({{\mathbb {T}}}\). Then the weakest topology that makes the t-characters \(\phi : G \rightarrow {{\mathfrak {T}}}\) continuous is called the Bohr-torsion topology on (G, t) and is denoted by \(t^\oplus \). When t is locally compact, we show that \(t^\oplus \) is Hausdorff if and only if (G, t) is zero dimensional, and if (G, t) is zero dimensional and H is a subgroup of G, then H is t-closed if and only if H is \(t^\oplus \)-closed.

We endow a topological group $(G, \tau)$ with a coarse structure defined by the smallest group ideal $S_{\tau} $ on $G$ containing all converging sequences with their limits and denote the obtained coarse group by $(G, S_{\tau})$. If $G$ is discrete then $(G, S_{\tau})$ is a finitary coarse group studding in Geometric Group Theory. The main result: if a topological abelian group $(G, \tau)$ contains a non-trivial converging sequence then $asdim \ (G, S_{\tau})= \infty $., Comment: Coarse structure, group ideal, asymptotic dimension, Hamming space. arXiv admin note: substantial text overlap with arXiv:1902.02320

We introduce, for any topological abelian group G, the property of equicontinuity of arcs of \(G^\wedge _p\), the dual group of G endowed with its pointwise topology. We analyze the implications of this property, which we denote by EAP\(_\sigma \), and we present some representative examples. Furthermore we prove that if G satisfies EAP\(_\sigma \), every element of the arcwise connected component of \(G^\wedge _p\) can be written as \(\phi (1)\) for a suitable one-parameter subgroup \(\phi :\mathbb {R} \rightarrow G^\wedge _p\).

Let E and F be topological vector spaces and let G and Y be topological abelian groups. We say that E is sequentially barrelled with respect to F if every sequence \((u_n)_{n\in \mathbb {N}}\) of continuous linear maps from E to F which converges pointwise to zero is equicontinuous. We say that G is barrelled with respect to F if every set \(\mathscr {H}\) of continuous homomorphisms from G to F, for which the set \( \mathscr {H}(x)\) is bounded in F for every \(x\in E\), is equicontinuous. Finally, we say that G is g-barrelled with respect to Y if every \(\mathscr {H}\subseteq \mathrm{CHom}(G,Y)\) which is compact in the product topology of \(Y^ G\) is equicontinuous. We prove that a barrelled normed space may not be sequentially barrelled with respect to a complete metrizable locally bounded topological vector space, a topological group which is a Baire space is barrelled with respect to any topological vector space, a topological group which is a Namioka space is g-barrelled with respect to any metrizable topological group, a protodiscrete topological abelian group which is a Baire space may not be g-barrelled (with respect to \(\mathbb R/\mathbb Z\)).

The notion of a weakly locally compact topological abelian group introduced in this paper generalizes the notion of a fibrous topological abelian group studied by N. Ya. Vilenkin.

This paper deals with the splitting of extensions of topological abelian groups. Given topological abelian groups G and H, we say that Ext ( G , H ) is trivial if every extension of topological abelian groups of the form 1 → H → X → G → 1 splits. We prove that Ext ( A ( Y ) , K ) is trivial for any free abelian topological group A ( Y ) over a zero-dimensional k ω -space Y and every compact abelian group K. Moreover we show that if K is a compact subgroup of a topological abelian group X such that the quotient group X / K is a zero-dimensional k ω -space, then there exists a continuous cross section from X / K to X. In the second part of the article we prove that Ext ( G , H ) is trivial whenever G is a product of locally precompact abelian groups and H has the form T α × R β for arbitrary cardinal numbers α and β. An analogous result is true if G = ∏ i ∈ I G i where each G i is a dense subgroup of a maximally almost periodic, Cech-complete group for which both Ext ( G i , R ) and Ext ( G i , T ) are trivial.

We study locally compact groups having all dense subgroups (locally) minimal. We call such groups densely (locally) minimal. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the groups Z p of p-adic integers. In [30] , we extended Prodanov's theorem to the non-abelian case at several levels. In this paper, we focus on the densely (locally) minimal abelian groups. We prove that in case that a topological abelian group G is either compact or connected locally compact, then G is densely locally minimal if and only if G either is a Lie group or has an open subgroup isomorphic to Z p for some prime p. This should be compared with the main result of [9] . Our Theorem C provides another extension of Prodanov's theorem: an infinite locally compact group is densely minimal if and only if it is isomorphic to Z p . In contrast, we show that there exists a densely minimal, compact, two-step nilpotent group that neither is a Lie group nor it has an open subgroup isomorphic to Z p .

A counterpart of the Mackey–Arens Theorem for the class of locally quasi-convex topological Abelian groups (LQC-groups) was initiated in Chasco et al. (Stud Math 132(3):257–284, 1999). Several authors have been interested in the problems posed there and have done clarifying contributions, although the main question of that source remains open. Some differences between the Mackey Theory for locally convex spaces and for locally quasi-convex groups, stem from the following fact: The supremum of all compatible locally quasi-convex topologies for a topological abelian group G may not coincide with the topology of uniform convergence on the weak quasi-convex compact subsets of the dual group $$G^\wedge $$ . Thus, a substantial part of the classical Mackey–Arens Theorem cannot be generalized to LQC-groups. Furthermore, the mentioned fact gives rise to a grading in the property of “being a Mackey group”, as defined and thoroughly studied in Diaz Nieto and Martin-Peinador (Proceedings in Mathematics and Statistics 80:119–144, 2014). At present it is not known—and this is the main open question—if the supremum of all the compatible locally quasi-convex topologies on a topological group is in fact a compatible topology. In the present paper we do a sort of historical review on the Mackey Theory, and we compare it in the two settings of locally convex spaces and of locally quasi-convex groups. We point out some general questions which are still open, under the name of Problems.

Anordinary Cech cohomology \( {\overset{\smile }{H}}^{\ast}\left(X,G\right) \) is defined for an arbitrary space X, and the group of coefficients G is assumed to be an Abelian group. On the category A C of compact pairs (X,A), an ordinary Cech cohomology satisfies the continuity axiom (see [1, Theorem 3.1.X]), i.e., we have the isomorphism $$ {\overset{\smile }{H}}^{*}\left(X,A,G\right)\approx \underrightarrow{ \lim }{\overset{\smile }{H}}^{*}\left({X}_m,{A}_m,G\right), $$ where \( \left(X,A\right)=\underleftarrow{ \lim}\left({X}_m,{A}_m\right),\left({X}_m,{A}_m\right)\in {A}_C \) Therefore, an ordinary Cech cohomology is called a continuous cohomology.

For a locally quasi-convex topological abelian group (G,τ), we study the poset (mathscr{C}(G,τ)) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G,(widehat{G})) . Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates ``from below'', our strategy consists of finding appropriate subgroups (H) of (G) that are easier to handle and show that (mathscr{C} (H)) and (mathscr{C} (G/H)) are large and embed, as a poset, in (mathscr{C}(G,τ)). Important special results are: (i) if (K) is a compact subgroup of a locally quasi-convex group (G), then (mathscr{C}(G)) and (mathscr{C}(G/K)) are quasi-isomorphic (3.15), (ii) if (D) is a discrete abelian group of infinite rank, then (mathscr{C}(D)) is quasi-isomorphic to the poset (mathfrak{F}_D) of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group (G ) with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset ( mathscr{C} (G) ) is as big as the underlying topological structure of (G,τ) (and set theory) allows. For a metrizable connected compact group (X), the group of null sequences (G=c_0(X)) with the topology of uniform convergence is studied. We prove that (mathscr{C}(G)) is quasi-isomorphic to (mathscr{P}(mathbb{R})) (6.9).

In this article we study the arcwise connected component in the Pontryagin dual of an abelian topological group. It is clear that the set of continuous characters that can be lifted over the reals is contained in the arcwise connected component of the dual group. We show that the converse is true if all arcs in the character group are equicontinuous sets. This property is present in Pontryagin duals of pseudocompact groups, of reflexive groups and of groups which are k -spaces as topological spaces. We study the meaning of such a property and its presence in groups and vector spaces endowed with weak topologies. We also characterize the image of the exponential mapping of a dual group as formed by those characters which can be lifted over the reals endowed with the Bohr topology.

In this paper, we study linearly topological groups. We introduce the notion of a weakly linearly compact group, which generalizes the notion of a weakly separable group, and examine the main properties of such groups. For weakly linearly compact groups, we construct the character theory and present an algebraic characterization of some classes of such groups. Some well-known theorems for periodic Abelian groups are generalized for the case of linearly discrete, topological Abelian groups; for linearly compact and linearly discrete topological Abelian groups, we also construct the character theory and study some important properties of linearly discrete groups. For linearly discrete, topological Abelian groups, we analyze the splittability condition (Theorem 3.12) and present the characteristic condition of decomposability of a discrete group G into the direct sum of rank-1 groups. We also present an algebraic characterization of linearly compact groups. We introduce the notion of a weakly linearly compact, topological Abelian group, which generalizes the notion of a weakly separable Abelian group, and examine some properties of such groups. These groups are a generalization of fibrous Abelian groups introduced by Vilenkin. We give an algebraic characterization of divisible, weakly locally compact Abelian groups that do not contain nonzero elements of finite order (Proposition 7.9). For weakly locally compact Abelian groups, we construct universal groups.

A Hausdorff topological group G is called lower continuous if the topology of G has no predecessor in G 2 ( G ) . The class of lower continuous topological groups contains all closed subgroups of products of minimal abelian groups, so strictly extend the class of minimal groups. Our main concern in this paper is the study of properties of lower continuous topological groups. Similar with the case for minimal groups, we provide a lower continuity criterion: a dense subgroup H of a Hausdorff topological abelian group G is lower continuous if and only if G is lower continuous and S o c ( G ) ≤ H . It is shown that every totally lower continuous abelian group is precompact. It is also shown that for a compact abelian groups G, G is hereditarily lower continuous if and only if G is torsion-free.

It has been claimed by Halmos in [Comment on the real line, Bull. Amer. Math. Soc., 50 (1944), 877-878] that if G is a Hausdorff locally compact topological abelian group and if the character group of G is torsion free then G is divisible. We prove that such claim is false, by presenting a family of counterexamples. While other counterexamples are known (see [D. L. Armacost, The structure of locally compact abelian groups, 1981]), we also present a family of stronger counterexamples, showing that even if one assumes that the character group of G is both torsion free and divisible, it does not follow that G is divisible., 5 pages, 0 figures

We prove that a topological group G is strongly countably complete (the notion introduced by Z. Frolík in 1961) iff G contains a closed countably compact subgroup H such that the quotient space G/H is completely metrizable and the canonical mapping π:G→G/H is closed. We also show that every strongly countably complete group is sequentially complete, has countable Gδ-tightness, and its completion is a Čech-complete topological group. Further, a pseudocompact strongly countably complete group is countably compact. An example of a pseudocompact topological Abelian group H with the Fréchet–Urysohn property is presented such that H fails to be sequentially complete, thus answering a question posed by Dikranjan, Martín Peinador, and Tarieladze in [Appl. Categor. Struct. 15 (2007) 511–539].

Let \({\mathcal{L}(X)}\) be the algebra of all bounded operators on a Banach space X. \({\theta:G\rightarrow \mathcal{L}(X)}\) denotes a strongly continuous representation of a topological abelian group G on X. Set \({\sigma^1(\theta(g)):=\{\lambda/|\lambda|,\lambda\in\sigma(\theta(g))\}}\), where σ(θ(g)) is the spectrum of θ(g) and \({\Sigma:=\{g\in G/\enskip\text{there is no} \enskip P\in \mathcal{P}/P\subseteq \sigma^1(\theta(g))\}}\), where \({\mathcal{P}}\) is the set of regular polygons of \({\mathbb{T}}\) (we call polygon in \({\mathbb{T}}\) the image by a rotation of a closed subgroup of \({\mathbb{T}}\), the unit circle of \({\mathbb{C}}\)). We prove here that if G is a locally compact and second countable abelian group, then θ is uniformly continuous if and only if Σ is non-meager.

Locally compact and weakly linearly compact topological groups are studied. The notion of a weakly linearly compact topological Abelian group is a generalization of the notion of a weakly separable topological Abelian group, introduced by N. Ya. Vilenkin. Some algebraic properties of these groups are studied.

In this paper, we prove an equivariant version of the classical Dold-Thom theorem. Associated to a finite group, a CW-complex on which this group acts and a covariant coefficient system in the sense of Bredon, we functorially construct a topological abelian group by the coend construction. Then we prove that the homotopy groups of this topological abelian group are naturally isomorphic to the Bredon equivariant homology of the CW-complex. At the end we present several examples of this result.

Given a Waelbroeck ring R, we prove that the Grassmannian of a projective finitely generated R -module is a topological manifold modeled on a topological abelian group of R -linear maps. Fibre bundles of fibre type a module as above, over a compact base space B, admitting R -valued partitions of unity, are classified by the homotopy classes of continuous maps on B with values in the respective Grassmannian. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

For a continuous surjection $T:X \to X$ on a compact metric space $X$ and a unimodular continuous weight on $X$, we consider a weighted composition operator $U_{T,w}$ on the Banach space $C(X)$ of complex-valued continuous functions on $X$ with the sup norm. The set ${\mathcal W}_{T}$ of all weights $w$, for which the operator $U_{T,w}$ has an eigenvalue with a unimodular eigenfunction, forms a topological abelian group. The group ${\mathcal W}_{T}$ admits a homomorphism $W_T$ to the first integral \v{C}ech cohomology of the space $X$. The image and the kernel of $W_T$ carry topological and ergodic aspects of the dynamics $T$. A concrete description of $\operatorname{Im}W_{T}$ and $\operatorname{Ker}W_{T}$ is given for positively expansive eventually-onto open maps (under an assumption on the induced homomorphism of the first \v{C}ech cohomology) and minimal rotations on tori.

The remainder of the completion of a topological abelian group (G, τ0) contains a nonzero element of prime order if and only if G admits a Hausdorff group topology τ1 that precedes the given topology and is such that (G, τ0) has no base of closed zero neighborhoods in (G, τ1).

We introduce a categorical closure operator g in the category of topological abelian groups (and continuous homomorphisms) as a Galois closure with respect to an appropriate Galois correspondence defined by means of the Pontryagin dual of the underlying group. We prove that a topological abelian group G is maximally almost periodic if and only if every cyclic subgroup of G is g-closed. This generalizes a property characterizing the circle group from (Studia Sci. Math. Hungar. 38 (2001) 97–113, A characterization of the circle group and the p-adic integers via sequential limit laws, preprint), and answers an appropriate version of a question posed in (A characterization of the circle group and the p-adic integers via sequential limit laws, preprint).

Throughout this abstract, G is a topological Abelian group and $$\hat G$$ is the space of continuous homomorphisms from G into the circle group $${\mathbb{T}}$$ in the compact-open topology. A dense subgroup D of G is said to determine G if the (necessarily continuous) surjective isomorphism $$\hat G \to \hat D$$ given by $$h \mapsto h\left| D \right.$$ is a homeomorphism, and G is determined if each dense subgroup of G determines G. The principal result in this area, obtained independently by L. Ausenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these. 1. There are (many) nonmetrizable, noncompact, determined groups. 2. If the dense subgroup D i determines G i with G i compact, then $$ \oplus _i D_i $$ determines Πi G i. In particular, if each G i is compact then $$ \oplus _i G_i $$ determines Πi G i. 3. Let G be a locally bounded group and let G + denote G with its Bohr topology. Then G is determined if and only if G + is determined. 4. Let non $$\left( {\mathcal{N}} \right)$$ be the least cardinal κ such that some $$X \subseteq {\mathbb{T}}$$ of cardinality κ has positive outer measure. No compact G with $$w\left( G \right) \geqslant non\left( {\mathcal{N}} \right)$$ is determined; thus if $$\left( {\mathcal{N}} \right) = {\mathfrak{N}}_1 $$ (in particular if CH holds), an infinite compact group G is determined if and only if w(G) = ω. Question. Is there in ZFC a cardinal κ such that a compact group G is determined if and only if w(G) < κ? Is $$\kappa = non\left( {\mathcal{N}} \right)?\kappa = {\mathfrak{N}}_1 ?$$

We prove that the asterisk topologies on the direct sum of topological Abelian groups, used by Kaplan and Banaszczyk in duality theory, are different. However, in the category of locally quasiconvex groups they do not differ, and coincide with the coproduct topology.  2003 Elsevier B.V. All rights reserved. MSC: 22A05

Let X ={ X ( t ), t ∈ T } be a stochastic process with independent increments indexed by the multidimensional set of parameters T=(R + ) q , q⩾1 , taking values in a T 0 topological Abelian group G . In this note, we give conditions under which the filtration F t =σ{X(s),s⩽t} , t ∈ T , generated by the process X is right continuous, i.e. F t =⋂ v>t F v for all t ∈ T .

We prove that every modular function on a multilattice L with values in a topological Abelian group generates a uniformity on L which makes the multilattice operations uniformly continuous with respect to the exponential uniformity on the power set of L.

For a topological abelian group X we topologize the group c 0 ( X ) $c_0(X)$ of all X-valued null sequences in a way such that when X = ℝ ${X={\mathbb {R}}}$ the topology of c 0 ( ℝ ) $c_0({\mathbb {R}})$ coincides with the usual Banach space topology of the classical Banach space c0 . If X is a non-trivial compact connected metrizable group, we prove that c 0 ( X ) $c_0(X)$ is a non-compact Polish locally quasi-convex group with countable dual group c 0 ( X ) ∧ $c_0(X)^{\wedge }$ . Surprisingly, for a compact metrizable X, countability of c 0 ( X ) ∧ $c_0(X)^{\wedge }$ leads to connectedness of X. Our principal application of the above results is to the class of locally quasi-convex Mackey groups ( LQC $\rm {LQC}$ -Mackey groups). A topological group ( G , μ ) $(G,\mu )$ from a class 𝒢 $\mathcal {G}$ of topological abelian groups will be called a Mackey group in 𝒢 $\mathcal {G}$ or a 𝒢 $\mathcal {G}$ -Mackey group if it has the following property: if ν is a group topology in G such that ( G , ν ) ∈ 𝒢 ${(G,\nu )\in \mathcal {G}}$ and ( G , ν ) $(G,\nu )$ has the same character group as ( G , μ ) $(G,\mu )$ , then ν ≤ μ ${\nu \le \mu }$ . Based upon the results obtained for c 0 ( X ) $c_0(X)$ , we provide a large family of metrizable precompact (hence, locally quasi-convex) connected groups which are not LQC $\rm {LQC}$ -Mackey. Namely, we show that for a connected compact metrizable group X ≠ { 0 } ${X\ne \lbrace 0\rbrace }$ , the group c 0 ( X ) $c_0(X)$ , endowed with the topology induced from the product topology on X ℕ $X^{{\mathbb {N}}}$ , is a metrizable precompact connected group which is not a Mackey group in LQC. Since metrizable locally convex spaces always carry the Mackey topology – a well-known fact from Functional Analysis –, our results prove that a Mackey theory for abelian groups is not a simple traslation of items known to hold for locally convex spaces. This paper is a contribution to the Mackey theory for groups, where properties of a topological nature like compactness or connectedness have an important role.

A topological Abelian group G is Pontryagin reflexive, or P-reflexive for short, if the natural homomorphism of G to its bidual group is a topological isomorphism. We look at the question, set by Kaplan in 1948, of characterizing the topological Abelian groups that are P-reflexive. Thus, we find some conditions on an arbitrary group G that are equivalent to the P-reflexivity of G and give an example that corrects a wrong statement appearing in previously existent characterizations of P-reflexive groups.

We define the g-extension of a topological Abelian group G as the set of all characters on G such that the restriction to every equicontinuous subset of G is continuous with respect to the pointwise convergence topology. A g-group is a topological Abelian group (G,τ) such that its g-extension coincides with its completion. The Bohr topology of a topological group (G,τ) is the topology that the group inherits as a subset of its Bohr compactification. A topological group (G,τ) respects a property P if the subsets A of G that satisfy the property P are exactly the same for the Bohr topology and for the original topology of the group [Trigos-Arrieta, J. Pure Appl. Algebra 70 (1991) 199]. All groups here are assumed to be Abelian. We prove that every complete g-group when endowed with its Bohr topology is a μ -space. As a consequence, we obtain that for a complete g-group the properties of respecting functionally boundedness, pseudocompactness, countable compactness and compactness are all equivalent and a characterization of this property is also provided. Finally, we extend a theorem of Rosenthal about the existence of sequences equivalent to the l 1 -basis. We prove that for a Cech-complete g-group the property of respecting compactness is equivalent to the existence of conveniently placed sequences equivalent to the l 1 -basis.

A topological abelian group G is P-reflexive if the natural homomorphism of G to its Pontryagin bidual group is a topological isomorphism. Let Cp(X) be the space of continuous functions with the topology of pointwise convergence. We investigate for what spaces X the group Cp(X) is P-reflexive. We show that: (1) if Cp(X) is P-reflexive, then X is a P-space; (2) there exists a non-discrete space X such that Cp(X) is P-reflexive; (3) there exists a P-space X such that Cp(X) is not P-reflexive; (4) there exists a simple space X for which the question of whether Cp(X) is P-reflexive is undecidable in ZFC.

We define GP-nuclear groups as topological Abelian groups for which the groups of summable and absolutely summable sequences are the same algebraically and topologically. It is shown that in the metrizable case only the algebraic coincidence of the mentioned groups is needed for GP-nuclearity. Some permanence properties of the class of GP-nuclear groups are obtained. Our final result asserts that nuclear groups in the sense of Banaszczyk are GP-nuclear. The validity of the converse assertion remains open.

We deal with the Krein-Langer problem for\(\mathcal{L}(\mathcal{H})\)-valued functions on the band (−2a, 2a)×Γ, where\(\mathcal{L}(\mathcal{H})\) is the algebra of continuous linear operators on a Hilbert space\(\mathcal{H}\),a a finite positive number and Γ a topological Abelian group. We show that every weakly continuous κ-indefinite function\(f:( - 2a,2a) \times \Gamma \to \mathcal{L}(\mathcal{H})\) admits a strongly continuous κ-indefinite continuation to ℝ × Γ with the same indefiniteness index κ. We give a parametrization of the extensions in terms of operator-valued Schur functions.