Second-countable space
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Feathered gyrogroups and gyrogroups with countable pseudocharacter Open
Topological gyrogroups, with a weaker algebraic structure than groups, have been investigated recently. In this paper, we prove that every feathered strongly topological gyrogroup is paracompact, which implies that every feathered strongly…
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LOCALLY NORMAL SUBGROUPS OF TOTALLY DISCONNECTED GROUPS. PART I: GENERAL THEORY Open
Let $G$ be a totally disconnected, locally compact group. A closed subgroup of $G$ is locally normal if its normalizer is open in $G$ . We begin an investigation of the structure of the family of closed locally normal subgroups of $G$ . Mo…
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Uniformly recurrent subgroups and the ideal structure of reduced crossed products Open
We study the ideal structure of reduced crossed product of topological dynamical systems of a countable discrete group. More concretely, for a compact Hausdorff space $X$ with an action of a countable discrete group $Γ$, we consider the ab…
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Around Property (T) for Quantum Groups Open
We study Property (T) for locally compact quantum groups, providing several\nnew characterisations, especially related to operator algebraic ergodic theory.\nQuantum Property (T) is described in terms of the existence of various Kazhdan\nt…
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Spaces with star countable extent Open
For a topological property $P$, we say that a space $X$ is star $P$ if for every open cover $\mathcal{U}$ of the space $X$ there exists $A\subset X$ such that $st (A,\mathcal{U})= X$. We consider space with star countable extent establishi…
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An uncountable Moore–Schmidt theorem Open
We prove an extension of the Moore–Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact…
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Rokhlin dimension for compact quantum group actions Open
We show that, for a given compact or discrete quantum group $G$, the class of actions of $G$ on C*-algebras is first-order axiomatizable in the logic for metric structures. As an application, we extend the notion of Rokhlin property for $G…
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Vanishing of $\ell ^2$-Betti numbers of locally compact groups as an invariant of coarse equivalence Open
We provide a proof that the vanishing of $\\ell^2$-Betti numbers of unimodular\nlocally compact second countable groups is an invariant of coarse equivalence.\n
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On rigid stabilizers and invariant random subgroups of groups of homeomorphisms Open
A generalization of the double commutator lemma for normal subgroups is shown for invariant random subgroups of a countable group acting faithfully on a Hausdorff space. As an application, we classify ergodic invariant random subgroups of …
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Uniformly locally compact and close to them spaces Open
In this work we study uniformly locally compact, uniformly locally Lindelöf and uniformly locally countably compact spaces. In particular, by using uniform structures, the paracompactness of any locally Lindelöf topological group and the c…
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Measure reducibility of countable Borel equivalence relations Open
We show that every basis for the countable Borel equivalence relations\nstrictly above $\\mathbb{E}_0$ under measure reducibility is uncountable,\nthereby ruling out natural generalizations of the Glimm-Effros dichotomy. We\nalso push many…
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Periodic measures are dense in invariant measures for residually finite amenable group actions with specification Open
We prove that for actions of a discrete countable residuallyfinite amenable group on a compact metric space with specification property, periodic measures are dense in theset of invariant measures. We also prove that certain expansiveactio…
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A Generalization of Strassen’s Theorem on Preordered Semirings Open
Given a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space tog…
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Uniform topology on EQ-algebras Open
In this paper, we use filters of an EQ -algebra E to induce a uniform structure ( E , 𝓚), and then the part 𝓚 induce a uniform topology 𝒯 in E . We prove that the pair ( E , 𝒯) is a topological EQ -algebra, and some properties of ( E , 𝒯) …
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On Polish groups admitting non-essentially countable actions Open
It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question po…
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Amenability and weak containment for actions of locally compact groups\n on $C^*$-algebras Open
In this work we introduce and study a new notion of amenability for actions\nof locally compact groups on $C^*$-algebras. Our definition extends the\ndefinition of amenability for actions of discrete groups due to Claire\nAnantharaman-Dela…
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Functional envelopes relative to the point-open topology on a subset Open
If $(X, f)$ is a dynamical system given by a locally compact separable metric space $X$ without isolated points and a continuous map $f : X\to X $ , and $A$ is a countable dense subset of $X$ , then by the functional envelope of $(X, f)$ r…
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Metrizability of Pseudo Topological Vector Spaces Open
In the present work, we introduce the notion of pseudo-seminorm, then we established, a criterion for the metrizability of a pseudo vector space 𝛯 with a pseudo topology 𝜏. It is specified, that for the metrizability of 𝛯 is necessary and …
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Normally preordered spaces and continuous multi-utilities Open
We study regular, normal and perfectly normal preorders by referring to suitable assumptions concerning the preorder and the topology of the space. We also present conditions for the existence of a countable continuous multi-utility repres…
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Invariant means and property $T$ of crossed products Open
Let $\\Gamma $ be a discrete group that acts on a semi-finite measure space $(\\Omega , \\mu )$ such that there is no $\\Gamma $-invariant function in $L^1(\\Omega , \\mu )$. We show that the absence of the $\\Gamma $-invariant mean on $L^…
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Analytic properties of approximate lattices Open
We introduce a notion of cocycle-induction for strong uniform approximate lattices in locally compact second countable groups and use it to relate (relative) Kazhdan- and Haagerup-type of approximate lattices to the corresponding propertie…
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On feebly compact topologies on the semigroup BF1ω Open
We study the Gutik-Mykhalenych semigroup B ω F 1 in the case when the family F 1 consists of the empty set and all singleton in ω . We show that B ω F 1 is isomorphic to subsemigroup B ω ↗ ω m i n of the Brandt ω -extension of the semilatt…
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Polish topologies on endomorphism monoids of relational structures Open
© 2023 Elsevier Inc. All rights reserved. This is the accepted manuscript version of an article which has been published in final form at https://doi.org/10.1016/j.aim.2023.109214
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On the semigroup B<sup>F</sup><sub>omega</sub> which is generated by the family F of atomic subsets of omega Open
We study the semigroup B ω F , which is introduced in [O. Gutik and M. Mykhalenych, On some generalization of the bicyclic monoid , Visnyk Lviv. Univ. Ser. Mech.-Mat. 90 (2020), 5-19], in the case when the family F of subsets of cardinalit…
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Almost elementariness and fiberwise amenability for étale groupoids Open
In this paper, we introduce two new approximation properties for étale groupoids, almost elementariness and (ubiquitous) fiberwise amenability, inspired by Matui's and Kerr's notions of almost finiteness. In fact, we show that, in their re…
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On spectra of Koopman, groupoid and quasi-regular representations Open
In this paper we investigate relations between Koopman, groupoid and quasi-regular representations of countable groups. We show that for an ergodic measure class preserving action of a countable group G on a standard Borel space the associ…
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Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group Open
We consider two twisted actions of a countable discrete group on $\sigma$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as i…
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Weak equivalence of stationary actions and the entropy realization problem Open
We introduce the notion of weak containment for stationary actions of a countable group and define a natural topology on the space of weak equivalence classes. We prove that Furstenberg entropy is an invariant of weak equivalence, and more…
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COMPUTABLE TOPOLOGICAL GROUPS Open
We investigate what it means for a (Hausdorff, second-countable) topological group to be computable. We compare several potential definitions based on classical notions in the literature. We relate these notions with the well-established d…
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Subsystem entropies of shifts of finite type and sofic shifts on countable amenable groups Open
In this work, we study the entropies of subsystems of shifts of finite type (SFTs) and sofic shifts on countable amenable groups. We prove that for any countable amenable group G , if X is a G -SFT with positive topological entropy $h(X)> …