Search

We examine sufficient and necessary conditions for the Choquet integral on the support of its integrand to be equal to the integral on a set which includes this support (we consider the problem in the case of a locally compact Hausdorff space).

We show that locally compact abelian p-groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. This result gives a new geometric interpretation for the classification of locally compact abelian groups which are rich in commuting closed subgroups. It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring. Among other things, we find applications for locally compact groups which are just noncompact.

We study in this paper conditions under which nilradicals of totally disconnected locally compact rings are closed. In the paper is given a characterization of locally finite compact rings via identities.

By computing the completely bounded norm of the flip map on the Haagerup tensor product $C_0 Y_1\otimes_{C_0 X} C_0 Y_2$ associated to a pair of continuous mappings of locally compact Hausdorff spaces $Y_1\rightarrow X\leftarrow Y_2$, we establish a simple characterisation of the Beck-Chevalley condition for base change of operator modules over commutative $C^*$-algebras, and a descent theorem for continuous fields of Hilbert spaces., Comment: 10 pages. Minor changes to the exposition; corrected an error in Remarks 6 not affecting the rest of the paper

A fuzzy compactum in a space X is an upper semi-continuous map f : A → [ 0 , 1 ] from a nonempty compact subspace A ⊂ X to the unit closed interval [ 0 , 1 ] such that { x ∈ A : f ( x ) > 0 } is dense in A. In this paper, we investigate the topological structure of the space K e ( X ) of fuzzy compacta of a separable metric space X with the Hausdorff metric. In particular, we prove that K e ( X ) ≈ l 2 if X is an infinite, locally compact, locally connected, connected and separable metric space.

For a given metric space X, we consider the set of all normal fuzzy sets on X, denoted by F ( X ) . In this work, we study expanding, positively expansive and weakly positively expansive dynamical systems ( X , f ) and how they are reflected in the dynamical system ( F ( X ) , f ˆ ) , where f ˆ is the Zadeh's extension of f and F ( X ) has one of the following metrics: the levelwise metric, the endograph metric, the sendograph metric and the Skorokhod metric. We mainly show that if we consider the following conditions: (i) ( X , f ) is positively expansive (resp. expanding); (ii) ( K ( X ) , f ‾ ) is positively expansive (resp. expanding); (iii) ( F ∞ ( X ) , f ˆ ) is positively expansive (resp. expanding); (iv) ( F 0 ( X ) , f ˆ ) is positively expansive (resp. expanding). Then (iv)⇒ (iii) ⇔ (ii) ⇒ (i). For expanding dynamical systems, we present a compact metric space and a locally compact metric space to show that (i) ⇏ (ii) and (iii) ⇏ (iv), respectively. For positively expansive dynamical systems, there is a compact metric space satisfying that (i) ⇏ (ii), but we don't know if (iii) ⇒ (iv).

A Polish group is said to be locally Roelcke precompact if there is a neighborhood of the identity element that is totally bounded in the Roelcke (or lower) group uniformity. These form a subclass of the locally bounded groups, while generalizing the Roelcke precompact and locally compact Polish groups. We characterize these groups in terms of their geometric structure as those locally bounded groups whose coarsely bounded sets are all Roelcke precompact, and in terms of their uniform structure as those groups whose completions in the Roelcke uniformity are locally compact. We also assess the conditions under which this locally compact space carries the structure of a semi-topological semigroup.

This paper has two goals: to present some new results that are necessary for further study and applications of quasi-linear functionals, and, by combining known and new results, to serve as a convenient single source for anyone interested in quasi-linear functionals on locally compact non-compact spaces or on compact spaces. We study signed and positive quasi-linear functionals paying close attention to singly generated subalgebras. The paper gives representation theorems for quasi-linear functionals on $C_c(X)$ and for bounded quasi-linear functionals on $C_0(X)$ on a locally compact space, and for quasi-linear functionals on $C(X)$ on a compact space. There is an order-preserving bijection between quasi-linear functionals and compact-finite topological measures, which is also "isometric" when topological measures are finite. Finally, we further study properties of quasi-linear functionals and give an explicit example of a quasi-linear functional., Comment: 30 pages

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.

In this paper, we investigate the homological notion of left $$\phi $$ -biprojectivity for certain Banach algebras, where $$\phi $$ is a nonzero multiplicative linear functional. Our initial result states that this notion is equivalent to left $$\phi $$ -contractibility, provided that $$\mathcal {A}$$ has a left approximate identity. As an application, we study left $$\phi $$ -biprojectivity of Banach algebras related to locally compact groups. For instance, we show that for a locally compact group $$\mathcal {G}$$ , the Segal algebra $$S(\mathcal {G})$$ is left $$\phi $$ -biprojective if and only if $$\mathcal {G}$$ is compact and the Fourier algebra $$A(\mathcal {G})$$ is left $$\phi $$ -biprojective if and only if $$\mathcal {G}$$ is discrete. Finally, we give some examples which show the differences between our new notion and the classical ones.

We introduce the notion of intrinsic semilattice entropy $\widetilde h$ in the category $\mathcal L_{qm}$ of generalized quasimetric semilattices and contractive homomorphisms. By using appropriate categories $\mathfrak X$ and functors $F:\mathfrak X\to\mathcal L_{qm}$ we find specific known entropies $\widetilde h_\mathfrak X$ on $\mathfrak X$ as intrinsic functorial entropies, that is, as $\widetilde h_\mathfrak X=\widetilde h\circ F$. These entropies are the intrinsic algebraic entropy, the algebraic and the topological entropies for locally linearly compact vector spaces, the topological entropy for locally compact totally disconnected groups and the algebraic entropy for locally compact compactly covered abelian groups.

The theories of metric spaces and fuzzy metric spaces are crucial topics in mathematics. Compactness is one of the most important and fundamental properties that have been widely used in Functional Analysis. In this paper, the definition of compact fuzzy soft metric space is introduced and some of its important theorems are investigated. Also, sequentially compact fuzzy soft metric space and locally compact fuzzy soft metric space are defined and the relationships between them are studied. Moreover, the relationships between each of the previous two concepts and several other known concepts are investigated separately. Besides, the compact fuzzy soft continuous functions are studied and some essential theorems are proved.

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.

It is known that if $f\colon {\mathbb R}^2 \to {\mathbb R}$ is a polynomial in each variable, then $f$ is a polynomial. We present generalizations of this fact, when ${\mathbb R}^2$ is replaced by $G\times H$, where $G$ and $H$ are topological Abelian groups. We show, e.g., that the conclusion holds (with generalized polynomials in place of polynomials) if $G$ is a connected Baire space and $H$ has a dense subgroup of finite rank or, for continuous functions, if $G$ and $H$ are connected Baire spaces. The condition of continuity can be omitted if $G$ and $H$ are locally compact or complete metric spaces. We present several examples showing that the results are not far from being optimal., Comment: 15 pages

Let $ {\mathcal{G}} $ be a locally compact groupoid. We show that there is a one-to-one correspondence between $ {\mathcal{G}} $ -spaces and the groupoid dynamical systems whose underling $ C_{0}({{\mathcal{G}}}^{(0)}) $ -algebra is commutative. We study minimality and (strong) proximality for $ {\mathcal{G}} $ -actions and show that each locally compact groupoid $ {\mathcal{G}} $ has a universal minimal (strongly) proximal $ {\mathcal{G}} $ -space (called the Furstenberg boundary).

In this paper, we characterize hypercyclic sequences of weighted translation operators on an Orlicz space in the context of locally compact hypergroups., 18 pages, Typos are corrected and one reference added

In this article, we study the existence and uniqueness of nonnegative solutions for the following parabolic problem{Htωu+∂u∂t=0, in X×(0,T),u(x,0)=u0(x),inX,where T > 0, X is a locally compact sepa...

We establish the boundedness of the Calderon-Zygmund operators on the Morrey spaces and the Hardy-Morrey spaces in locally compact Vilenkin groups.

Using our results from [4], we present a new proof of the extension of the Stone duality theorem to the category BooleSp of zero-dimensional locally compact Hausdorff spaces (= Boolean spaces) and continuous maps that was provided by Dimov [3], and obtain two new duality theorems for the category BooleSp.

It is shown that a general concept of Morita duality between abelian categories with no generating hypothesis for reflexive objects is completely described by a special class of quasi-abelian categories, called ample Morita categories. The duality takes place between a pair of intrinsic abelian full subcategories which exist for any quasi-abelian category. Morita categories, being slightly more general, admit a natural embedding into ample ones. An existence criterion for a duality of a Morita category is proved. It generalizes Pontrjagin duality for the category of locally compact abelian groups which is shown to be a non-ample non-classical Morita category. More examples of non-classical Morita categories are obtained from dual systems of topological vector spaces satisfying the Hahn-Banach property.

Some results between the properties of strongly topological gyrogroups and the properties of their remainders are established. In particular, if a strongly topological gyrogroup G is non-locally compact and G has a first-countable remainder, then $$\chi (G)\le \omega _{1}$$ , $$\omega (G)\le 2^{\omega }$$ and $$|bG|\le 2^{\omega _{1}}$$ . Moreover, it is proved that the property of paracompact p-space of a strongly topological gyrogroup G is equivalent with G having a Lindelof remainder in a compactification. By this result, we prove that if H is a dense subspace of a strongly topological gyrogroup G which is locally pseudocompact and not locally compact, then every remainder of H is pseudocompact. Furthermore, if a strongly topological gyrogroup G has countable pseudocharacter and G is non-metrizable, then all remainders of G are pseudocompact. These two results give partial answers to a question posed by Arhangel’ skiǐ and Choban, see (Topol Appl 157:789–799, 2010, Problem 5.1). Finally, it is shown that the Lindelof property of a non-locally compact strongly topological gyrogroup G is equivalent with having a remainder with subcountable type for some compactifications of G.

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 $\Gamma$.

We give conditions for boundedness of Hausdorff operators on real Hardy spaces H1 over homogeneous spaces of locally compact groups with local doubling property. Special cases of the hyperbolic plane and 2-sphere are considered. In this cases Hausdorff operators look as orbital like integrals.

Two non-discrete Hausdorff group topologies τ and δ on a group G are called transversal if the least upper bound τ ⋁ δ of τ and δ is the discrete topology. In this paper, we discuss the existence of transversal group topologies on locally pseudocompact, locally precompact, or locally compact groups. We prove that each locally pseudocompact, connected topological group satisfies central subgroup paradigm, which gives an affirmative answer to a problem posed by Dikranjan, Tkachenko, and Yaschenko [Topology Appl., 2006, 153: 3338–3354]. For a compact normal subgroup K of a locally compact totally disconnected group G, if G admits a transversal group topology, then G/K admits a transversal group topology, which gives a partial answer again to a problem posed by Dikranjan, Tkachenko, and Yaschenko in 2006. Moreover, we characterize some classes of locally compact groups that admit transversal group topologies.

Amenable actions of locally compact groups on von Neumann algebras are investigated by exploiting the natural module structure of the crossed product over the Fourier algebra of the acting group. The resulting characterisation of injectivity for crossed products generalises a result of Anantharaman-Delaroche on discrete groups. Amenable actions of locally compact groups on $C^*$-algebras are investigated in the same way, and amenability of the action is related to nuclearity of the corresponding crossed product. A survey is given to show that this notion of amenable action for $C^*$-algebras satisfies a number of expected properties. A notion of inner amenability for actions of locally compact groups is introduced, and a number of applications are given in the form of averaging arguments, relating approximation properties of crossed product von Neumann algebras to properties of the components of the underlying $w^*$-dynamical system. We use these results to answer a recent question of Buss-Echterhoff-Willett., Comment: Fixed a mistake in Section 6. Typo fixes

Let G be a locally compact group and let ${\mathcal {SUB}(G)}$ be the hyperspace of closed subgroups of G endowed with the Chabauty topology. The main purpose of this paper is to characterise the connectedness of the Chabauty space ${\mathcal {SUB}(G)}$ . More precisely, we show that if G is a connected pronilpotent group, then ${\mathcal {SUB}(G)}$ is connected if and only if G contains a closed subgroup topologically isomorphic to ${{\mathbb R}}$ .

Let $F/F_{\mathsf{o}}$ be a quadratic extension of non-archimedean locally compact fields of odd residual characteristic and $\sigma$ be its non-trivial automorphism. We show that any $\sigma$-self-dual cuspidal representation of ${\rm GL}_n(F)$ contains a $\sigma$-self-dual Bushnell--Kutzko type. Using such a type, we construct an explicit test vector for Flicker's local Asai $L$-function of a ${\rm GL}_n(F_{\mathsf{o}})$-distinguished cuspidal representation and compute the associated Asai root number. Finally, by using global methods, we compare this root number to Langlands--Shahidi's local Asai root number, and more generally we compare the corresponding epsilon factors for any cuspidal representation., Comment: 61 pages, final version to appear in JEMS

We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find $L_{\omega_1 \omega}$-axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan's property {\bf (T)} can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures., Comment: 15 pages; to appear in Communications in Mathematics, special issue "Ostrava seminar on Mathematical Physics". arXiv admin note: substantial text overlap with arXiv:1706.04157