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S. S. Gabriyelyan

Publications and source records attributed to S. S. Gabriyelyan.

16 recordsLinked to original sources
arXiv2017

On the Ascoli property for locally convex spaces and topological groups

We characterize Ascoli spaces by showing that a Tychonoff space $X$ is Ascoli iff the canonical map from the free locally convex space $L(X)$ over $X$ into $C_k\big(C_k(X)\big)$ is an embedding of locally convex spaces. We prove that an uncountable direct sum of non-trivial locally convex spaces is not Ascoli. If a $c_0$-barrelled space $X$ is weakly Ascoli, then $X$ is linearly isomorphic to a dense subspace of $\mathbb{R}^Γ$ for some $Γ$. Consequently, a Fréchet space $E$ is weakly Ascoli iff $E=\mathbb{R}^N$ for some $N\leqω$. If $X$ is a $μ$-space and a $k$-space (for example, metrizable), then $C_k(X)$ is weakly Ascoli iff $X$ is discrete. We prove that the weak* dual space of a Banach space $E$ is Ascoli iff $E$ is finite-dimensional.

math.GN↗
arXiv2015

An Open Mapping Theorem

It is proved that any surjective morphism $f: \mathbb{Z}^κ\to K$ onto a locally compact group $K$ is open for every cardinal $κ$. This answers a question posed by Karl Heinrich Hofmann and the second author.

math.GN↗
arXiv2014

On $\mathfrak{P}$-spaces and related concepts

The concept of the strong Pytkeev property, recently introduced by Tsaban and Zdomskyy in [32], was successfully applied to the study of the space $C_c(X)$ of all continuous real-valued functions with the compact-open topology on some classes of topological spaces $X$ including Čech-complete Lindelöf spaces. Being motivated also by several results providing various concepts of networks we introduce the class of $\mathfrak{P}$-spaces strictly included in the class of $\aleph$-spaces. This class of generalized metric spaces is closed under taking subspaces, topological sums and countable products and any space from this class has countable tightness. Every $\mathfrak{P}$-space $X$ has the strong Pytkeev property. The main result of the present paper states that if $X$ is an $\aleph_0$-space and $Y$ is a $\mathfrak{P}$-space, then the function space $C_c(X,Y)$ has the strong Pytkeev property. This implies that for a separable metrizable space $X$ and a metrizable topological group $G$ the space $C_c(X,G)$ is metrizable if and only if it is Fréchet-Urysohn. We show that a locally precompact group $G$ is a $\mathfrak{P}$-space if and only if $G$ is metrizable.

math.GN↗
arXiv2014

On topological spaces and topological groups with certain local countable networks

Being motivated by the study of the space $C_c(X)$ of all continuous real-valued functions on a Tychonoff space $X$ with the compact-open topology, we introduced in [15] the concepts of a $cp$-network and a $cn$-network (at a point $x$) in $X$. In the present paper we describe the topology of $X$ admitting a countable $cp$- or $cn$-network at a point $x\in X$. This description applies to provide new results about the strong Pytkeev property, already well recognized and applicable concept originally introduced by Tsaban and Zdomskyy [43]. We show that a Baire topological group $G$ is metrizable if and only if $G$ has the strong Pytkeev property. We prove also that a topological group $G$ has a countable $cp$-network if and only if $G$ is separable and has a countable $cp$-network at the unit. As an application we show, among the others, that the space $D'(Ω)$ of distributions over open $Ω\subseteq\mathbb{R}^{n}$ has a countable $cp$-network, which essentially improves the well known fact stating that $D'(Ω)$ has countable tightness. We show that, if $X$ is an $\mathcal{MK}_ω$-space, then the free topological group $F(X)$ and the free locally convex space $L(X)$ have a countable \mbox{$cp$-network}. We prove that a topological vector space $E$ is $p$-normed (for some \mbox{$0<p\leq 1$}) if and only if $E$ is Fréchet-Urysohn and admits a fundamental sequence of bounded sets.

math.GN↗
arXiv2013

On topological properties of the group of the null sequences valued in an Abelian topological group

Following [23], denote by $\mathfrak{F}_0$ the functor on the category $\mathbf{TAG}$ of all Hausdorff Abelian topological groups and continuous homomorphisms which passes each $X\in \mathbf{TAG}$ to the group of all $X$-valued null sequences endowed with the uniform topology. We prove that if $X\in \mathbf{TAG}$ is an $(E)$-space (respectively, a strictly angelic space or a Š-space), then $\mathfrak{F}_0 (X)$ is an $(E)$-space (respectively, a strictly angelic space or a Š-space). We study respected properties for topological groups in particular from categorical point of view. Using this investigation we show that for a locally compact Abelian (LCA) group $X$ the following are equivalent: 1) $X$ is totally disconnected, 2) $\mathfrak{F}_0 (X)$ is a Schwartz group, 3) $\mathfrak{F}_0 (X)$ respects compactness, 4) $\mathfrak{F}_0(X)$ has the Schur property. So, if a LCA group $X$ has non-zero connected component, the group $\mathfrak{F}_0(X)$ is a reflexive non-Schwartz group which does not have the Schur property. We prove also that for every compact connected metrizable Abelian group $X$ the group $\mathfrak{F}_0 (X)$ is monothetic that generalizes a result by Rolewicz for $X=\mathbb{T}$.

math.GR↗
arXiv2012

Topologies on groups determined by sets of convergent sequences

A Hausdorff topological group $(G,τ)$ is called an $s$-group and $τ$ is called an $s$-topology if there is a set $S$ of sequences in $G$ such that $τ$ is the finest Hausdorff group topology on $G$ in which every sequence of $S$ converges to the unit. The class $\mathbf{S}$ of all $s$-groups contains all sequential Hausdorff groups and it is finitely multiplicative. A quotient group of an $s$-group is an $s$-group. For a non-discrete topological group $(G,τ)$ the following three assertions are equivalent: 1) $(G,τ)$ is an $s$-group, 2) $(G,τ)$ is a quotient group of a Graev free topological group over a metrizable space, 3) $(G,τ)$ is a quotient group of a Graev free topological group over a sequential Tychonoff space. The Abelian version of this characterization of $s$-groups holds as well.

math.GR↗
arXiv2012

Characterizing sequences for precompact group topologies

A precompact group topology $τ$ on an abelian group $G$ is called {\em single sequence characterized} (for short, {\em ss-characterized}) if there is a sequence $\mathbf{u}= (u_n)$ in $G$ such that $τ$ is the finest precompact group topology on $G$ making $\mathbf{u}=(u_n)$ converge to zero. It is proved that a metrizable precompact abelian group $(G,τ)$ is $ss$-characterized iff it is countable. For every metrizable precompact group topology $τ$ on a countably infinite abelian group $G$ there exists a group topology $η$ such that $η$ is strictly finer than $τ$ and the groups $(G,τ)$ and $(G,η)$ have the equal Pontryagin dual groups. We give a complete description of all $ss$-characterized precompact abelian groups modulo countable $ss$-characterized groups from which we derive: (1) No infinite pseudocompact abelian group is $ss$-characterized. (2) An $ss$-characterized precompact abelian group is hereditarily disconnected.

math.GR↗
arXiv2011

Minimally almost periodic group topology on infinite countable Abelian groups: A solution to Comfort's question

For any countable subgroup $H$ of an unbounded Abelian group $G$ there is a complete Hausdorff group topology $τ$ such that $H$ is the von Neumann radical of $(G,τ)$. In particular, we obtain the positive answer to Comfort's question: any unbounded countable Abelian group admits a complete Hausdorff minimally almost periodic (MinAP) group topology. A bounded infinite Abelian group admits a MinAP group topology if and only if all its leading Ulm-Kaplansky invariants are infinite. If, in addition, $G$ is countably infinite, a MinAP group topology can be chosen to be complete.

math.GR↗
arXiv2011

Absolute continuity and singularity of two probability measures on a filtered space

Let $μ$ and $ν$ be fixed probability measures on a filtered space $(Ω, {\cal F}, ({\cal F}_t)_{t\in {\bf R}^{+}})$. Denote by $μ_T $ and $ν_T $ (respectively, $μ_{T-} $ and $ν_{T-} $) the restrictions of the measures $μ$ and $ν$ on ${\cal F}_T $ (respectively, on ${\cal F}_{T-} $) for a stopping time $T$. We find the Hahn decomposition of $μ_T $ and $ν_T $ using the Hahn decomposition of the measures $μ$, $ν$, and the Hellinger process $h_t$ in the strict sense of order 1/2. The norm of the absolutely continuous component of $μ_{T-} $ with respect to $ν_{T-} $ is computed in terms of density processes and Hellinger integrals.

math.PR↗
arXiv2011

Pontryagin duality for Abelian $s$- and $sb$-groups

The main goal of the article is to study the Pontryagin duality for Abelian $s$- and $sb$-groups. Let $G$ be an infinite Abelian group and $X$ be the dual group of the discrete group $G_d$. We show that a dense subgroup $H$ of $X$ is $\mathfrak{g}$-closed iff $H$ algebraically is the dual group of $G$ endowed with some maximally almost periodic $s$-topology. Every reflexive Polish Abelian group is $\mathfrak{g}$-closed in its Bohr compactification. If a $s$-topology $τ$ on a countably infinite Abelian group $G$ is generated by a countable set of convergent sequences, then the dual group of $(G,τ)$ is Polish. A non-trivial Hausdorff Abelian topological group is a $s$-group iff it is a quotient group of the $s$-sum of a family of copies of $(\mathbb{Z}^\mathbb{N}_0, \mathbf{e})$.

math.GR↗
arXiv2010

Minimally almost periodic group topology on countable torsion Abelian groups

For any countable torsion subgroup $H$ of an unbounded Abelian group $G$ there is a complete Hausdorff group topology $τ$ such that $H$ is the von Neumann radical of $(G,τ)$. In particular, any unbounded torsion countable Abelian group admits a complete Hausdorff minimally almost periodic (MinAP) group topology. If $G$ is a bounded torsion countably infinite Abelian group, then it admits a MinAP group topology if and only if all its leading Ulm-Kaplansky invariants are infinite. In such a case, a MinAP group topology can be chosen to be complete.

math.GR↗
arXiv2009

Groups of quasi-invariance and the Pontryagin duality

A Polish group $G$ is called a group of quasi-invariance or a QI-group, if there exist a locally compact group $X$ and a probability measure $μ$ on $X$ such that 1) there exists a continuous monomorphism of $G$ to $X$, and 2) for each $g\in X$ either $g\in G$ and the shift $μ_g$ is equivalent to $μ$ or $g\not\in G$ and $μ_g$ is orthogonal to $μ$. It is proved that $G$ is a $σ$-compact subset of $X$. We show that there exists a quotient group $\mathbb{T}^H_2$ of $\ell^2$ modulo a discrete subgroup which is a Polish monothetic non locally quasi-convex (and hence nonreflexive) pathwise connected QI-group, and such that the bidual of $\mathbb{T}^H_2$ is not a QI-group. It is proved also that the bidual group of a QI-group may be not a saturated subgroup of $X$.

math.GN↗
arXiv2009

On $T$-sequences and characterized subgroups

Let $X$ be a compact metrizable abelian group and $\mathbf{u}=\{u_n\}$ be a sequence in its dual $X^{\wedge}$. Set $s_{\mathbf{u}} (X)= \{x: (u_n,x)\to 1\}$ and $\mathbb{T}_0^H = \{(z_n)\in \mathbb{T}^{\infty} : z_n\to 1 \}$. Let $G$ be a subgroup of $X$. We prove that $G=s_{\mathbf{u}} (X)$ for some $\mathbf{u}$ iff it can be represented as some dually closed subgroup $G_{\mathbf{u}}$ of ${\rm Cl}_X G \times \mathbb{T}_0^H$. In particular, $s_{\mathbf{u}} (X)$ is polishable. Let $\mathbf{u}=\{u_n\}$ be a $T$-sequence. Denote by $(\widehat{X}, \mathbf{u})$ the group $X^{\wedge}$ equipped with the finest group topology in which $u_n \to 0$. It is proved that $(\widehat{X}, \mathbf{u})^{\wedge} =G_{\mathbf{u}}$ and $\mathbf{n} (\widehat{X}, \mathbf{u}) = s_{\mathbf{u}} (X)^{\perp}$. We also prove that the group generated by a Kronecker set can not be characterized.

math.GN↗
arXiv2009

Characterization of almost maximally almost-periodic groups

Let $G$ be an abelian group. We prove that a group $G$ admits a Hausdorff group topology $τ$ such that the von Neumann radical $\mathbf{n}(G, τ)$ of $(G, τ)$ is non-trivial and finite iff $G$ has a non-trivial finite subgroup. If $G$ is a topological group, then $\mathbf{n} (\mathbf{n} (G)) \not= \mathbf{n} (G)$ if and only if $\mathbf{n} (G)$ is not dually embedded. In particular, $\mathbf{n} (\mathbf{n} (\mathbb{Z},τ)) = \mathbf{n} (\mathbb{Z},τ)$ for any Hausdorff group topology $τ$ on $\mathbb{Z}$.

math.GN↗