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We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost finite principal groupoids lacking other desirable properties, such as amenability or even a-T-menability. This behaviour is in stark contrast to the case of principal transformation groupoids associated to group actions., Comment: Revised version. To appear in Ergodic Theory and Dynamical Systems

For a second countable locally compact abelian group G, we study a system of translates generated by $$f \in L^2 (G)$$ . We find some equivalent conditions of this family to have some fundamental frame properties. More precisely, let $$\Gamma $$ be a uniform lattice in G (a closed subgroup which is cocompact and discrete) and $$\Gamma ^*$$ be the annihilator of $$\Gamma $$ in $${\widehat{G}}$$ . For $$f \in L^2(G)$$ , the $$\Gamma ^*$$ -periodic function $$\Phi _f$$ is defined as $$\Phi _f (\xi ) = \sum _{\gamma \in {\Gamma } ^*} | {\widehat{f}} (\xi + \gamma ) |^2$$ on $${\widehat{\Gamma }}$$ (the dual group of $$\Gamma $$ ) and some of its properties are investigated. In particular, it is shown that if $$\Phi _f$$ is continuous, then the family $$\lbrace f(.+ \gamma ) \rbrace _{\gamma \in \Gamma }$$ cannot be a redundant frame. Among other things, it is shown that there is an isometry from $$L^2(G)$$ into $$L^2({\widehat{\Gamma }})$$ in such a way that the system of translates in $$L^2({\widehat{\Gamma }})$$ is transferred to a nice Fourier-like system in $$L^2({\widehat{\Gamma }})$$ . Also, the canonical and oblique duals of the frames of translates are investigated.

Let G be a locally compact second countable Hausdorff space with a positive regular Borel measure and ω is a weight on G. In this article, we provide necessary and sufficient conditions for the hyp...

Let $\Gamma$ be an amenable countable discrete group. Fix an ergodic free nonsingular action of $\Gamma$ on a nonatomic standard probability space. Let $G$ be a compactly generated locally compact second countable group such that the closure of the group of inner automorphisms of $G$ is compact in the natural topology. It is shown that there exists a {\it bounded} ergodic $G$-valued cocycle of $\Gamma$.

For a separable finite diffuse measure space $${\mathcal {M}}$$ M and an orthonormal basis $$\{\varphi _n\}$$ { φ n } of $$L^2({\mathcal {M}})$$ L 2 ( M ) consisting of bounded functions $$\varphi _n\in L^\infty ({\mathcal {M}})$$ φ n ∈ L ∞ ( M ) , we find a measurable subset $$E\subset {\mathcal {M}}$$ E ⊂ M of arbitrarily small complement $$|{\mathcal {M}}{\setminus } E| | M \ E | < ϵ , such that every measurable function $$f\in L^1({\mathcal {M}})$$ f ∈ L 1 ( M ) has an approximant $$g\in L^1({\mathcal {M}})$$ g ∈ L 1 ( M ) with $$g=f$$ g = f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of $${\mathcal {M}}=G/H$$ M = G / H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.

Let $\mathcal {G}$ be a second countable, Hausdorff topological group. If $\mathcal {G}$ is locally compact, totally disconnected and T is an expansive automorphism then it is shown that the dynamical system $(\mathcal {G}, T)$ is topologically conjugate to the product of a symbolic full-shift on a finite number of symbols, a totally wandering, countable-state Markov shift and a permutation of a countable coset space of $\mathcal {G}$ that fixes the defining subgroup. In particular if the automorphism is transitive then $\mathcal {G}$ is compact and $(\mathcal {G}, T)$ is topologically conjugate to a full-shift on a finite number of symbols.

We classify the locally compact second-countable (l.c.s.c.) groups $A$ that are abelian and topologically characteristically simple. All such groups $A$ occur as the monolith of some soluble l.c.s.c. group $G$ of derived length at most $3$; with known exceptions (specifically, when $A$ is $\mathbb{Q}^n$ or its dual for some $n \in \mathbb{N}$), we can take $G$ to be compactly generated. This amounts to a classification of the possible isomorphism types of abelian chief factors of l.c.s.c. groups, which is of particular interest for the theory of compactly generated locally compact groups., Comment: 14 pages

We study Borel homomorphisms $\theta : G\rightarrow H$ for arbitrary locally compact second countable groups $G$ and $H$ for which the measure $$\theta_*(\mu )(\alpha )=\mu (\theta ^{-1}(\alpha ))\quad \text{for } \quad \alpha \subseteq H $$ is absolutely continuous with respect to $\nu,$ where $\mu $ (resp. $\nu $) is a Haar measure for $G,$ (resp. $H$). We define a natural mapping $\mathcal G$ from the class of maximal abelian selfadjoint algebra bimodules (masa bimodules) in $B(L^2(H))$ into the class of masa bimodules in $B(L^2(G))$ and we use it to prove that if $k\subseteq G\times G$ is a set of operator synthesis, then $(\theta \times \theta)^{-1} (k)$ is also a set of operator synthesis and if $E\subseteq H$ is a set of local synthesis for the Fourier algebra $A(H)$, then $\theta ^{-1}(E)\subseteq G$ is a set of local synthesis for $A(G).$ We also prove that if $\theta ^{-1}(E)$ is an $M$-set (resp. $M_1$-set), then $E$ is an $M$-set (resp. $M_1$-set) and if $Bim(I^\bot )$ is the masa bimodule generated by the annihilator of the ideal $I$ in $VN(G)$, then there exists an ideal $J$ such that $\mathcal G(Bim(I^\bot ))=Bim(J^\bot ).$ If this ideal $J$ is an ideal of multiplicity then $I$ is an ideal of multiplicity. In case $\theta_*(\mu )$ is a Haar measure for $\theta (G)$ we show that $J$ is equal to the ideal $\rho_*(I)$ generated by $\rho (I),$ where $\rho (u)=u\circ \theta , \;\;\forall \;u\;\in \;I.$

In this paper we show that for an almost finite minimal ample groupoid G, its reduced $$C^*$$ -algebra $$C_r^*(G)$$ has real rank zero and strict comparison even though $$C_r^*(G)$$ may not be nuclear in general. Moreover, if we further assume G being also second countable and non-elementary, then its Cuntz semigroup $${\mathrm{Cu}}(C_r^*(G))$$ is almost divisible and $${\mathrm{Cu}}(C_r^*(G))$$ and $${\mathrm{Cu}}(C_r^*(G)\otimes {\mathcal {Z}})$$ are canonically order-isomorphic, where $${\mathcal {Z}}$$ denotes the Jiang-Su algebra.

We give a new proof of the Baum--Connes conjecture with coefficients for any second countable, locally compact topological group that acts properly and cocompactly on a finite-dimensional CAT(0)-cubical space with bounded geometry. The proof uses the Julg-Valette complex of a CAT(0)-cubical space introduced by the first three authors, and the direct splitting method in Kasparov theory developed by the last author., 25 pages

espanolNuestro proposito es presentar una exposicion elemental de unresultado clasico en topologia general, que es una version debil de un problemaconocido como la conjetura del espacio normal de Moore. Con este fin,se definen y estudian algunas propiedades basicas de los espacios de Moorey se caracterizan aquellos que a su vez son segundo contables y de Lindelof.Ademas, se enuncia la hipotesis del continuo y el Axioma de Martin, y seaplican para establecer la independencia del problema en cuestion. EnglishOur purpose is to present an elementary exposition of a classicalresult in general topology which is a weak version of a problem known as thenormal Moore space conjecture. With this aim we study some of the basicproperties of Moore spaces and characterize those which are both Lindelofand second countable. We also make use of the continuum hypothesis alongwith Martin’s axiom to establish the result in question.

Let X be a second countable locally compact Abelian group. We prove some group analogues of the Skitovich–Darmois, Heyde and Kac–Bernstein characterization theorems for Q -independent random variables taking values in the group X .

The class of weakly linearly Lindelof spaces was introduced and studied by Juhasz, Tkachuk and Wilson in [7] . Recall that a space X is weakly linearly Lindelof if for any family U of non-empty open subsets of X of regular uncountable cardinality κ, there exists a point x ∈ X such that every neighborhood of x meets κ-many elements of U . In this paper, we show that: (1) If X is a weakly linearly Lindelof space and U is an open cover of X, then for the open cover { St 2 ( x , U ) : x ∈ X } of X, there exists a countable subset A ⊂ X such that St 2 ( A , U ) ‾ = X ; (2) Every weakly linearly Lindelof normal metaLindelof space is weakly Lindelof; (3) If X is a first countable regular space, then M ( X ) (generated by Moore Machine) is weakly linearly Lindelof if and only if X is weakly linearly Lindelof; (4) Every product of a weakly linearly Lindelof space and a space of countable spread (or a separable space) is weakly linearly Lindelof; (5) If a subspace X ⊂ ω 1 ω is weakly linearly Lindelof, then X is second countable (and hence, metrizable); (6) If X is a weakly linearly Lindelof Baire space with a rank 2-diagonal such that w e ( X ) ≤ ω 1 , then | X | ≤ c ; (7) The space X is cellular-WLL if and only if it is weakly linearly Lindelof.

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-C*-algebra, introduced by Barlak, Szabó, and Voigt in the case when G is second countable and coexact, to an arbitrary compact quantum group G. All the the preservations and rigidity results for Rokhlin actions of second countable coexact compact quantum groups obtained by Barlak, Szabó, and Voigt are shown to hold in this general context. As a further application, we extend the notion of equivariant order zero dimension for equivariant *-homomorphisms, introduced in the classical setting by the first and third authors, to actions of compact quantum groups. This allows us to define the Rokhlin dimension of an action of a compact quantum group on a C*-algebra, recovering the Rokhlin property as Rokhlin dimension zero. We conclude by establishing a preservation result for finite nuclear dimension and finite decomposition rank when passing to fixed point algebras and crossed products by compact quantum group actions with finite Rokhlin dimension.

Let G be a second countable locally compact group with type-I left regular representation, G ˆ its dual and K = ( K π ) π ∈ G ˆ a specific measurable field of operators. In this paper, we investigate an inversion formula for L p ( G ) . Let 1 p , r ≤ 2 , 1 p + 1 q = 1 s + 1 r = 1 , and F p : L p ∩ L 1 ( G ) ⟶ L q ( G ˆ ) be defined by F p ( f ) π = π ( f ) K π 1 q . The map F p extends uniquely to a linear map of L p ( G ) into L q ( G ˆ ) , denoted by F p . Let F ¯ p be the transpose of F p and f ∈ L p ( G ) . We prove that f ¯ ∈ F ¯ r [ L r ( G ˆ ) ] if and only if F p ( f ) K 1 p − 1 s ∈ L r ( G ˆ ) , and, if that is the case, we have f ¯ = F ¯ r ( K 1 p − 1 s [ F p ( f ) ] ⁎ ) .

We establish that an uncountable space X must be essentially uncountable whenever its extent and tightness are countable. As a consequence, the equality $$\mathrm{ext}(X)= t(X)=\omega $$ implies that the space $$C_{p}(X, [0,1])$$ is discretely selective. If X is a metrizable space, then $$C_{p}(X, [0,1])$$ has the Banakh property if and only if so does $$C_{p}(Y, [0,1])$$ for some closed separable $$Y\subset X$$. We apply the above results to show that, for a metrizable X, the space $$C_{p}(X, [0,1])$$ is strongly dominated by a second countable space if and only if X is homeomorphic to $$D\,{\oplus }\, M$$ where D is a discrete space and M is countable. For a metrizable space X, we also prove that $$C_{p}(X,[0,1])$$ has the Lindelof $$\Sigma $$-property if and only if the set of non-isolated points of X is second countable. Our results solve several open questions.

Dana P. Williams raised in [Proc. Am. Math. Soc., Ser. B, 2016] the following question: Must a second countable, locally compact, transitive groupoid have open range map? This paper gives a negative answer to that question. Although a second countable, locally compact transitive groupoid G may fail to have open range map, we prove that we can replace its topology with a topology which is also second countable, locally compact, and with respect to which G is a topological groupoid whose range map is open. Moreover, the two topologies generate the same Borel structure and coincide on the fibres of G., Comment: 7 pages

Having potential applications in multiplexing techniques and in the synthesis of frames, orthogonality (or strongly disjointness) plays a significant role in frame theory (e.g. construction of new frames from existing ones, constructions related with duality, etc.). In this article, we study orthogonality of a pair of frames over locally compact abelian (LCA) groups. We start with the investigation of the dual Gramian analysis tools of Ron and Shen through a pre-Gramian operator over the set-up of LCA groups. Then we fiberize some operators associated with Bessel families generated by unitary actions of co-compact (not necessarily discrete) subgroups of LCA groups. Using this fiberization, we study and characterize a pair of orthogonal frames generated by the action of a unitary representation ρ of a co-compact subgroup Γ ⊂ G on a separable Hilbert space L 2 ( G ) , where G is a second countable LCA group. Precisely, we consider frames of the form { ρ ( γ ) ψ : γ ∈ Γ , ψ ∈ Ψ } for a countable family Ψ in L 2 ( G ) . We pay special attention to this problem in the context of translation-invariant space by assuming ρ as the action of Γ on L 2 ( G ) by left-translation. The representation of Γ acting on L 2 ( G ) by (left-)translation is called the (left-)regular representation of Γ. Further, we apply our results on co-compact Gabor systems over LCA groups. At this juncture, it is pertinent to note that the resulting characterization can be useful for constructing new frames by using various techniques including the unitary extension principle by Ron and Shen [24] and its recent extension to LCA groups by Christensen and Goh [7] .

Let X be an arbitrary set. A topology t on X is said to be useful if every complete and continuous preorder on X is representable by a continuous real-valued order preserving function. It will be shown, in a first step, that there exists a natural one-to-one correspondence between continuous and complete preorders and complete separable systems on X . This result allows us to present a simple characterization of useful topologies t on X . According to such a characterization, a topology t on X is useful if and only if for every complete separable system E on ( X , t ) the topology t E generated by E and by all the sets X ∖ E ¯ is second countable. Finally, we provide a simple proof of the fact that the countable weak separability condition (cwsc), which is closely related to the countable chain condition (ccc), is necessary for the usefulness of a topology.

Let G be a second countable locally compact abelian group. The aim of this paper is to characterize the left translates on the Heisenberg group H ( G ) to be frames and Riesz bases in terms of the group Fourier transform.

v4 is the final version; International audience; In an earlier paper we established that every second countable, completely normal spectral space is homeomorphic to the ℓ-spectrum of some Abelian ℓ-group. We extend that result to ℓ-spectra of vector lattices over any countable totally ordered division ring k. Extending our original machinery, about finite lattices of polyhedra, from linear to affine and allowing relativizations to convex subsets, then invoking Baro's Normal Triangulation Theorem, we obtain the following result:Theorem. For every countable formally real field k, every second countable, completely normal spectral space is homeomorphic to the real spectrum of some commutative unital k-algebra.The countability assumption on k is necessary: there exists a second countable, completely normal spectral space that cannot be embedded, as a spectral subspace, into either the ℓ-spectrum of any right vector lattice over an uncountable directed partially ordered division ring, or the real spectrum of any commutative unital algebra over an uncountable field.

Let $\G$ be a unimodular type I second countable locally compact group and $\wG$ its unitary dual. Motivated by a recent pseudo-differential calculus, we develop a positive Berezin-type quantization with operator-valued symbols defined on $\wG\times\G$., 16 pages

Let E be Hausdorff locally compact second countable spaces (HLCSC) and $$(2^{E}, 2^{f})$$ (hit-or-miss topology equipped) be hyperspace dynamical system induced by a given dynamical system (E, f). In this paper, the concepts of topologically co-compact ergodicity (resp. topologically co-compact strong ergodicity) and topologically co-compact double ergodicity (resp. topologically co-compact double strong ergodicity) are introduced for dynamical systems. For any HLCSC system (E, f), these three conditions on (E, f) are, respectively, equivalent to topological ergodicity (resp. topologically strong ergodicity) and topological double ergodicity (resp. topological double strong ergodicity) on $$(2^{E}, 2^{f})$$. The concept of topologically co-compact exact (c-exact) is also introduced, and we show that if f is perfect and c-exact, then $$2^{f}:{\mathcal {F}}_{00}\rightarrow {\mathcal {F}}_{00}$$ is topologically exact, where $${\mathcal {F}}_{00}=\{F\in {\mathcal {F}}_{0}:$$F is finite$$\}$$ and $${\mathcal {F}}_{0}=2^{E}$$. In addition, other noticeable properties of topologically co-compact ergodicity (resp. topologically co-compact strong ergodicity) and topologically co-compact double ergodicity (resp. topologically co-compact double strong ergodicity) are studied.

We study fluctuations of ergodic averages generated by actions of amenable groups. In the setting of an abstract ergodic theorem for locally compact second countable amenable groups acting on uniformly convex Banach spaces, we deduce a highly uniform bound on the number of fluctuations of the ergodic average for a class of F{\o}lner sequences satisfying an analogue of Lindenstrauss's temperedness condition. Equivalently, we deduce a uniform bound on the number of fluctuations over long distances for arbitrary F{\o}lner sequences. As a corollary, these results imply associated bounds for a continuous action of an amenable group on a $\sigma$-finite $L^{p}$ space with $p\in(1,\infty)$., Comment: 14 pages. Journal article version of results previously appearing in the thesis arXiv:1901.08538 (with some minor corrections). To appear in Bull. London Math. Soc

This article is concerned with measure equivalence and uniform measure equivalence of locally compact, second countable groups. We show that two unimodular, locally compact, second countable groups are measure equivalent if and only if they admit free, ergodic, probability measure preserving actions whose cross section equivalence relations are stably orbit equivalent. Using this we prove that in the presence of amenability any two such groups are measure equivalent and that both amenability and property (T) are preserved under measure equivalence, extending results of Connes-Feldman-Weiss and Furman. Furthermore, we introduce a notion of uniform measure equivalence for unimodular, locally compact, second countable groups, and prove that under the additional assumption of amenability this notion coincides with coarse equivalence, generalizing results of Shalom and Sauer. Throughout the article we rigorously treat measure theoretic issues arising in the setting of non-discrete groups., Comment: v2: results significantly expanded and many new results added. SR added as coauthor. v3: typos fixed, slight change in title, added Corollary C and Theorem E. v4: the definition of UME has been slightly changed, correcting a mistake in v3. v5: minor changes; to appear in Groups, Geometry, and Dynamics

Let $${\mathcal {G}}$$ be a topological locally compact Hausdorff and second countable groupoid with a Haar system and $${\mathcal {K}}$$ a proper subgroupoid of $${\mathcal {G}}$$ with a Haar system too. $$({\mathcal {G}},{\mathcal {K}})$$ is an internally Gelfand pair if for any u in the unit space, the pair of isotropy groups $$({\mathcal {G}}(u), {\mathcal {K}}(u))$$ is a Gelfand pair. In this work, we prove, when $$({\mathcal {G}},{\mathcal {K}})$$ is an internally Gelfand pair that any representation of $${\mathcal {C}}^{*} ({\mathcal {G}}, {\mathcal {K}},\lambda , \alpha )$$ induced from a positive definite spherical function on an isotropy group is irreducible. We introduce also the notion of spherical bundle and after identifying it with the spectrum of some $$C^*$$ -algebra, we obtain some properties of its topology.

For a Young function $$\phi $$ and a locally compact second countable group G, let $$L^\phi (G)$$ denote the Orlicz space on G. In this paper, we present a necessary and sufficient condition for the topological transitivity of a sequence of cosine operators $$\{C_n\}_{n=1}^{\infty }:=\{\frac{1}{2}(T^n_{g,w}+S^n_{g,w})\}_{n=1}^{\infty }$$ , defined on $$L^{\phi }(G)$$ . We investigate the conditions for a sequence of cosine operators to be topologically mixing. Further, we go on to prove a similar result for the direct sum of a sequence of cosine operators. Finally, we give an example of topologically transitive sequence of cosine operators.

A Hausdorff topological space X is called superconnected (resp. coregular) if for any nonempty open sets U 1 , … U n ⊆ X , the intersection of their closures U ‾ 1 ∩ … ∩ U ‾ n is not empty (resp. the complement X ∖ ( U ‾ 1 ∩ … ∩ U ‾ n ) is a regular topological space). A canonical example of a coregular superconnected space is the projective space Q P ∞ of the topological vector space Q ω = { ( x n ) n ∈ ω ∈ Q ω : | { n ∈ ω : x n ≠ 0 } | ω } over the field of rationals Q . The space Q P ∞ is the quotient space of Q ω ∖ { 0 } ω by the equivalence relation x ∼ y iff Q ⋅ x = Q ⋅ y . We prove that every countable second-countable coregular space is homeomorphic to a subspace of Q P ∞ , and a topological space X is homeomorphic to Q P ∞ if and only if X is countable, second-countable, and admits a decreasing sequence of closed sets ( X n ) n ∈ ω such that (i) X 0 = X , ⋂ n ∈ ω X n = ∅ , (ii) for every n ∈ ω and a nonempty relatively open set U ⊆ X n the closure U ‾ contains some set X m , and (iii) for every n ∈ ω the complement X ∖ X n is a regular topological space. Using this topological characterization of Q P ∞ we find topological copies of the space Q P ∞ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.

Let $\Delta$ be a closed, cocompact subgroup of $G \times \widehat{G}$, where $G$ is a second countable, locally compact abelian group. Using localization of Hilbert $C^*$-modules, we show that the Heisenberg module $\mathcal{E}_{\Delta}(G)$ over the twisted group $C^*$-algebra $C^*(\Delta,c)$ due to Rieffel can be continuously and densely embedded into the Hilbert space $L^2(G)$. This allows us to characterize a finite set of generators for $\mathcal{E}_{\Delta}(G)$ as exactly the generators of multi-window (continuous) Gabor frames over $\Delta$, a result which was previously known only for a dense subspace of $\mathcal{E}_{\Delta}(G)$. We show that $\mathcal{E}_{\Delta}(G)$ as a function space satisfies two properties that make it eligible for time-frequency analysis: Its elements satisfy the fundamental identity of Gabor analysis if $\Delta$ is a lattice, and their associated frame operators corresponding to $\Delta$ are bounded., Comment: 24 pages; several changes have been made to the presentation, while the content remains essentially unchanged; to appear in Journal of Fourier Analysis and Applications

In this paper, we prove the existence of a particular diagonalization for normal bounded operators defined on subspaces of $$L^2({\mathfrak {S}})$$ where $${\mathfrak {S}}$$ is a second countable LCA group. The subspaces where the operators act are invariant under the action of a group $$\Gamma $$ which is a semi-direct product of a uniform lattice of $${\mathfrak {S}}$$ with a discrete group of automorphisms. This class includes the crystal groups which are important in applications as models for images. The operators are assumed to be $$\Gamma $$ preserving. i.e. they commute with the action of $$\Gamma $$ . In particular we obtain a spectral decomposition for these operators. This generalizes recent results on shift-preserving operators acting on lattice invariant subspaces where $${\mathfrak {S}}$$ is the Euclidean space.

In this paper we prove the existence of a time-frequency space that best approximates a given finite set of data. Here best approximation is in the least square sense, among all time-frequency spaces with no more than a prescribed number of generators. We provide a formula to construct the generators from the data and give the exact error of approximation. The setting is in the space of square integrable functions defined on a second countable LCA group and we use the Zak transform as the main tool.

In this note we prove that if X is a Tychonoff space and $$X^2 {\setminus }\Delta $$ is dominated by a second countable space then X is cosmic. This solves an open problem of Cascales et al. (Topol Appl 158:204–214). We also consider the case when X is compact and $$X^2 {\setminus }\Delta $$ is dominated by a metric space M; in this situation we show that if such domination is strong, then the tightness of X is bounded by the weight of M.

The goals of this paper are twofold. First, we generalize the result of Gaboriau and Lyons [GL07] to the setting of von Neumann's problem for equivalence relations, proving that for any non-amenable ergodic probability measure preserving (pmp) equivalence relation $\mathcal{R}$, the Bernoulli extension over a non-atomic base space $(K, \kappa)$ contains the orbit equivalence relation of a free ergodic pmp action of $\mathbb{F}_2$. Moreover, we provide conditions which imply that this holds for any non-trivial probability space $K$. Second, we use this result to prove that any non-amenable unimodular locally compact second countable group admits uncountably many free ergodic pmp actions which are pairwise not von Neumann equivalent (hence, pairwise not orbit equivalent).

In this paper, we give some properties of rectifiable spaces and their relationship with P-space, metrizable space. These results are used to generalize some results in [2], [9] and [12]. Moreover, we give the conditions for a rectifiable space to be second-countable.

In this paper, we show that if the reduced Fourier–Stieltjes algebra B ρ ( G ) of a second countable locally compact group G has either weak* fixed point property or asymptotic center property, then G is compact. As a result, we give affirmative answers to open problems raised by Fendler and et al. in 2013. We then conclude that a second countable group with a discrete reduced dual must be compact. This generalizes a theorem of Baggett. We also construct a compact scattered Hausdorff space Ω for which the dual of the scattered C*-algebra C ( Ω ) lacks weak* fixed point property. The C*-algebra C ( Ω ) provides a negative answer to a question of Randrianantoanina in 2010. In addition, we prove a variant of Bruck's generalized fixed point theorem for the preduals of von Neumann algebras. Furthermore, we give some examples emphasizing that the conditions in Bruck's generalized conjecture (BGC) can not be weakened any more.

In this article we show that a contra second countable bitopological space is a $p_1$-Lindelof space, but the converse is not true in general. We provide suitable example with the help of concepts of nest and interlocking from LOTS. The relation between pairwise regular spaces and $p_1$-normal spaces is studied. At the end, we propose some open questions which may enrich various concepts related to Lindelofness in a bitopological space and other areas of mathematical ideas.

In this article, we provide necessary and sufficient conditions for the orthogonality of two Bessel families when such families have the form of generalized translation invariant (GTI) systems over a second countable locally compact abelian (LCA) group G. The work is motivated by the utility of a recent notion given by Jakobsen and Lemvig on GTI systems in L 2 ( G ) , and the concept of the orthogonality (or strongly disjointness) of a pair of frames studied by Balan, Han, and Larson. Consequently, we deduce similar results for several function systems including the case of TI systems, and GTI systems on compact abelian groups. We apply our results to the Bessel families having wave-packet structure (combination of wavelet as well as Gabor structure), and hence a characterization for pairwise orthogonal wave-packet frame systems over LCA groups is obtained. In addition, we relate the well-established theory from literature with our results by observing several deductions in the context of wavelet and Gabor systems over LCA groups with G = R d , Z d , etc.

We study uniform and non-uniform model sets in arbitrary locally compact second countable (lcsc) groups, which provide a natural generalization of uniform model sets in locally compact abelian groups as defined by Meyer and used as mathematical models of quasi-crystals. We then define a notion of auto-correlation for subsets of finite local complexitiy in arbitrary lcsc groups, which generalizes Hof's classical definition beyond the class of amenable groups, and prov ide a formula for the auto-correlation of a regular model set. Along the way we show that the punctured hull of an arbitrary regular model set admits a unique invariant probability measure, even in the case where the punctured hull is non-compact and the group is non-amenable. In fact this measure is also the unique stationary measure with respect to any admissible probability measure.

The Zak transform on $\mathbb{R}^d$ is an important tool in condensed matter physics, signal processing, time-frequency analysis, and harmonic analysis in general. This article introduces a generalization of the Zak transform to a class of locally compact $G$-spaces, where $G$ is either a locally compact abelian or a second countable unimodular type I group. This framework unifies previously proposed generalizations of the Zak transform. It is shown that the Zak transform has invariance properties analog to the classic case and is a Hilbert space isomorphism between the space of $L^2$-functions and a direct integral of Hilbert spaces that is explicitly determined via a Weil formula for $G$-spaces and a Poisson summation formula for compact subgroups. Some applications in physics are outlined.

A space X is weakly linearly Lindelöf if for any family U of non-empty open subsets of X of regular uncountable cardinality κ, there exists a point x ∈ X such that every neighborhood of x meets κ-many elements of U. We also introduce the concept of almost discretely Lindelöf spaces as the ones in which every discrete subspace can be covered by a Lindelöf subspace. We prove that, in addition to linearly Lindelöf spaces, both weakly Lindelöf spaces and almost discretely Lindelöf spaces are weakly linearly Lindelöf. The main result of the paper is formulated in the title. It implies that every weakly Lindelöf monotonically normal space is Lindelöf, a result obtained earlier in [3]. We show that, under the hypothesis 2ω < ωω, if the co-diagonal ΔcX = (X × X) \ΔX is discretely Lindelöf, then X is Lindelöf and has a weaker second countable topology; here ΔX = {(x, x): x ∈ X} is the diagonal of the space X. Moreover, discrete Lindelöfness of ΔcX together with the Lindelöf Σ-property of X imply that X has a countable network.

Given a Tychonoff space X, let F ( X ) and A ( X ) be respectively the free topological group and the free Abelian topological group over X in the sense of Markov. For every n ∈ N , let F n ( X ) (resp. A n ( X ) ) denote the subspace of F ( X ) (resp. A ( X ) ) that consists of words of reduced length at most n with respect to the free basis X. In this paper, we mainly discuss the subspaces F n ( X ) and A n ( X ) with countable tightness for a Lasnev space X, and prove that: (1) Assume b = ω 1 . For a non-metrizable Lasnev space X, the tightness of F 5 ( X ) is countable if and only if the tightness of F ( X ) is countable; (2) Let X be the closed image of a locally separable metrizable space. Then the tightness of A 4 ( X ) is countable if and only if the tightness of A ( X ) is countable.

We establish that a Tychonoff space X is countable if and only if C p ( X ) is strongly dominated by a second countable space. The same is true for a compact space K such that C p ( K , [ 0 , 1 ] ) is strongly dominated by a second countable space. We also prove that strong domination by a second countable space of the complement of the diagonal of a Tychonoff space X implies that X is an ℵ 0 -space. Our results solve several published open questions.

We give an explicit description of the tracial state simplex of the $C^*$-algebra $C^*(G)$ of an arbitrary connected, second countable, locally compact, solvable group $G$. We show that every tracial state of $C^*(G)$ lifts from a tracial state of the $C^*$-algebra of the abelianized group, and the intersection of the kernels of all the tracial states of $C^*(G)$ is a proper ideal unless $G$ is abelian. As a consequence, the $C^*$-algebra of a connected solvable nonabelian Lie group cannot embed into a simple unital AF-algebra., Comment: 9 pages

In the absence of the Axiom of Choice, necessary and sufficient conditions for a locally compact Hausdorff space to have all non-empty second-countable compact Hausdorff spaces as remainders are given in ZF. Among other independence results, the characterization of locally compact Hausdorff spaces having all non-empty metrizable compact spaces as remainders, obtained by Hatzenbuhler and Mattson in ZFC, is proved to be independent of ZF. Urysohn's Metrization Theorem is generalized. New concepts of a strongly filterbase infinite set and a dyadically filterbase infinite set are introduced, both stemming from the investigations on compactifications. Set-theoretic and topological definitions of the new concepts are given, and their relationship with certain known notions of infinite sets is investigated in ZF. A new permutation model is introduced in which there exists a strongly filterbase infinite set which is weakly Dedekind-finite.

We bound the nuclear dimension of crossed products associated to some partial actions of finite groups or $\mathbb{Z}$ on finite dimensional locally compact Hausdorff second countable spaces. Our results apply to globalizable partial actions, finite group partial actions, minimal partial automorphisms, and partial automorphisms acting on zero-dimensional spaces, or a class of one dimensional spaces, containing $1$-dimensional CW complexes. This extends work on global systems by Hirshberg and Wu., Comment: 25 pages, 1 figure. To appear in J. Funct. Anal. Minor changes, accepted version

We establish that a monolithic compact space X is not scattered if and only if has a dense subset without non-trivial convergent sequences. Besides, for any cardinal $$\kappa \geqslant \mathfrak {c}$$ , the space $$\mathbb {R}^\kappa $$ has a dense subspace without non-trivial convergent sequences. If X is an uncountable $$\sigma $$ -compact space of countable weight, then any dense set has a dense subspace without non-trivial convergent sequences. We also prove that for any countably compact sequential space X, if has a dense k-subspace, then X is scattered.

We show that, for a Lindelof Σ-space X, if C p ( X , [ 0 , 1 ] ) is strongly dominated by a second countable space, then X is countable. Under Martin's Axiom we prove that there exists a countable space Z that strongly dominates the complement of the diagonal of any first countable compact space. In particular, strong domination by a countable space of the complement of the diagonal of a compact space X need not imply metrizability of X. It turns out that the same countable space Z strongly dominates C p ( X ) for an uncountable space X. Our results solve several published open problems.

For each finite set S of prime numbers there exists a unique completion ℚ S of ℚ, which is a second countable, locally compact and totally disconnected topological ring. This topological ring has a natural ultrametric that allows to define a pseudodifferential operator D α and to study an abstract heat equation on the Hilbert space L 2(ℚ S ). The fundamental solution of this equation is a normal transition function of a Markov process on ℚ S . The techniques developed provides a general framework for these kind of problems on different ultrametric groups.