Topological Group Research Articles

Weakly first-countability in strongly topological gyrogroups

In this note, it is proved that (1) if (G,τ,⊕) is a strongly topological gyrogroup and H is a closed strong subgyrogroup of G, then G/H is κ-Fréchet-Urysohn if and only if G/H is strongly κ-Fréchet-Urysohn under the condition that H is neutral; (2) let H be a closed strong subgyrogroup of a strongly topological gyrogroup(G,τ,⊕), then the equality Δ(G/H)=ψ(G/H) holds when H is neutral; (3) if (G,τ,⊕) is a sequential strongly topological gyrogroup having a point-countable k-network, then G is metrizable or a topological sum of cosmic subspaces. There results improve the related results in topological groups.

Representations of groups on Banach spaces

We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let G \mathcal {G} be a topological group, and A \mathcal {A} a unital symmetric C ∗ C^* -subalgebra of U C ( G ) \mathrm {UC}(\mathcal {G}) , the algebra of bounded uniformly continuous functions on G \mathcal {G} . Generalizing the notion of a stable metric, we study A \mathcal {A} -metrics δ \delta , i.e., the function δ ( e , ⋅ ) \delta (e, \cdot ) belongs to A \mathcal {A} ; the case A = W A P ( G ) \mathcal {A}=W\hskip -0.7mm A\hskip -0.2mm P(\mathcal {G}) , the algebra of weakly almost periodic functions on G \mathcal {G} , recovers stability. If the topology of G G is induced by a left invariant metric d d , we prove that A \mathcal {A} determines the topology of G \mathcal {G} if and only if d d is uniformly equivalent to a left invariant A \mathcal {A} -metric. As an application, we show that the additive group of C [ 0 , 1 ] C[0,1] is not reflexively representable; this is a new proof of Megrelishvili [Topological transformation groups: selected topics, Elsevier, 2007, Question 6.7] (the problem was already solved by Ferri and Galindo [Studia Math. 193 (2009), pp. 99–108] with different methods and later the results were generalized by Yaacov, Berenstein, and Ferri [Math. Z. 267 (2011), pp.129–138]). Let now G \mathcal {G} be a metric group, and assume A ⊆ L U C ( G ) \mathcal {A}\subseteq \mathrm {LUC}(\mathcal {G}) , the algebra of bounded left uniformly continuous functions on G \mathcal {G} , is a unital C ∗ C^* -algebra which is the uniform closure of coefficients of representations of G \mathcal {G} on members of F \mathscr {F} , where F \mathscr {F} is a class of Banach spaces closed under ℓ 2 \ell _2 -direct sums. We prove that A \mathcal {A} determines the topology of G \mathcal {G} if and only if G \mathcal {G} embeds into the isometry group of a member of F \mathscr {F} , equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability.

Numerical studies of the energy absorption capacities and deformation mechanisms of 2D cellular topologies

This paper investigates the energy absorption capacities of selected cellular topologies under quasi-static loading conditions. Twenty topologies with nearly identical relative densities belonging to 4 groups were examined: honeycomb, re-entrant, bioinspired and chiral. The topologies were modeled using an experimentally validated numerical ABSplus model and subsequently subjected to in-plane uniaxial compression tests. The findings revealed the topologies with the most favorable energy absorption parameters and the main deformation mechanisms. The topologies were classified by mechanism, and a parametric study of basic material properties, namely modulus of elasticity, yield stress, and ductility, was performed for a representative topology from each mechanism. The results indicated that the honeycomb group topologies were characterized by the largest average absorbed energy, and yield stress was found to have the greatest impact on energy absorption efficiency regardless of the main deformation mechanism.

Proximal groups: Extension of topological groups. Application in the concise representation of Hilbert envelopes on oscillatory motion waveforms

In this paper, we introduce proximal groups that are a generalization of topological groups. A straight-forward application of proximal groups is given in the concise representation of collections of overlapping Hilbert envelope lobes attached to the peak points on oscillatory motion waveforms that occur in sequences of video frames that track object movements.

On topological groups of monotonic automorphisms

We study topological groups of monotonic automorphisms on a generalized ordered space L. We find a condition that is necessary and sufficient for the set of all monotonic automorphism on L along with the function composition and the topology of point-wise convergence to be a topological group.

Partial actions on quotient spaces and globalization

Given a partial action of a topological group G on a space X we determine properties P which can be extended from X to its globalization. We treat the cases when P is any of the following: Hausdorff, regular, metrizable, second countable, and having invariantmetric. Further, for a normal subgroup H, we introduce and study a partial action of G/H on the orbit space of X; applications to invariant metrics and inverse limits are presented.

On absolutely C-embedded topological groups

A topological group H is called absolutely C-embedded if H is C-embedded into every topological group containing H as a subgroup. It is shown that a left uniformly continuous real-valued function on a subgroup of a topological group G can extend to a continuous function on G. Therefore, fine groups are absolutely C-embedded. We also prove that a CM-factorizable groups are absolutely C-embedded. Additionally, we explore the factorizability of absolutely C-embedded groups and establish that an absolutely C-embedded group G is R-factorizable if and only if G is an ω-narrow group. We also show that a non-metrizable absolutely C-embedded group G is M-factorizable if and only if G is τ-precompact, where τ=Fi(G).

Measure-theoretic equicontinuity and rigidity of group actions

Let ( G , X ) be a G-system, which means that X is a compact Hausdorff space and G is an infinite topological group continuously acting on X, and let μ be a G-invariant measure of ( G , X ) . In this paper, we introduce the concepts of rigidity, uniform rigidity and μ-Ω-equicontinuity of ( G , X ) with respect to an infinite sequence Ω of G and the notions of μ-Ω-equicontinuity and μ-Ω-mean-equicontinuity of a function f ∈ L 2 ( μ ) with respect to an infinite sequence Ω of G. Then we give some equivalent conditions for f ∈ L 2 ( μ ) and ( G , X ) to be rigid, respectively. In addition, if G is commutative and X satisfies the first axiom of countability, we present some equivalent conditions for ( G , X ) to be uniformly rigid.

Tensor products of topological abelian groups and Pontryagin duality

Let G be the group of all Z-valued homomorphisms of the Baer-Specker group ZN. The group G is algebraically isomorphic to Z(N), the infinite direct sum of the group of integers, and equipped with the topology of pointwise convergence on ZN, becomes a non reflexive prodiscrete group. It was an open question to find its dual group Gˆ. Here, we answer this question by proving that Gˆ is topologically isomorphic to ZN⊗QT, the (locally quasi-convex) tensor product of ZN and T. Furthermore, we investigate the reflexivity properties of the groups Cp(X,Z), the group of all Z-valued continuous functions on X equipped with the pointwise convergence topology, and Ap(X), the free abelian group on a 0-dimensional space X equipped with the topology tp(C(X,Z)) of pointwise convergence topology on C(X,Z). In particular, we prove that Ap(X)ˆ≃Cp(X,Z)⊗QT and we establish the existence of 0-dimensional spaces X such that Cp(X,Z) is Pontryagin reflexive.

Fixed Point Set and Equivariant Map of a S-Topological Transformation Group

The fixed point set and equivariant map of a S-topological transformation group is explored in this work. For any subset K of G, it is established that the fixed point set XK is clopen in X and for a free S-topological transformation group, it is proved that the fixed point set of K is equal to the fixed point set of closure and interior of the subgroup of G generated by K. Subsequently, it is proved that the map between STCG(X) and STCG’(X’) is a homomorphism under a Φ’- equivariant map. Also, it is proved that there is an isomorphism between the quotient topological groups and some basic properties of fixed point set of a S-topological transformation group are studied.

On the relative L-theory and the relative signature of PL manifolds with boundary

In this paper, we give a new description of the relative topological structure group of PL manifolds with boundary, and put the relative L-group into an exact sequence of groups. Then we define the relative index of PL manifolds with boundary, and show that this relative index will induce an additive map from the relative L-group to the K-theory of the relative Roe algebra.

Constructing Number Field Isomorphisms from *-Isomorphisms of Certain Crossed Product C*-Algebras

We prove that the class of crossed product C*-algebras associated with the action of the multiplicative group of a number field on its ring of finite adeles is rigid in the following explicit sense: Given any *-isomorphism between two such C*-algebras, we construct an isomorphism between the underlying number fields. As an application, we prove an analogue of the Neukirch–Uchida theorem using topological full groups, which gives a new class of discrete groups associated with number fields whose abstract isomorphism class completely characterises the number field.

On a New Type of Topological Transformation Group

On a New Type of Topological Transformation Group

Properties of Hemicompact Topological Groups

Properties of Hemicompact Topological Groups

Group topologies on automorphism groups of homogeneous structures

Group topologies on automorphism groups of homogeneous structures

TOPOLOGICAL MODULES GEOMETRY

The extremal structure of zero-neighbourhoods of a topological module is analyzed reaching unexpected conclusions when the module topology is not Hausdorff. These results motivate us to introduce the notion of metric modules, which are modules endowed with a translation-invariant metric, turning them into an (additive) topological group. We study the central and diametral points of additively symmetric subsets and find examples of convex sets which are not symmetric translates (translates of addivitely symmetric subsets). As a consequence of all of these, it seems natural to transport the well-known Bishop-Phelps property from the category of real topological vector spaces to general topological modules over topological rings. Then we stick to particular topological rings, the unital C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras, showing that the subset of positive elements lying below the unity is an effect algebra. We also prove that every continuous linear operator on a Hausdorff locally convex topological vector space that commutes with all continuous linear projections of one-dimensional range is a multiple of the identity. Finally, we discuss how to transport the previous result to C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras.

Quotient and Product of Center Topological Groups

Quotient and Product of Center Topological Groups

Reflexive group topologies on the integers generated by sequences

We establish reflexivity of a family of group topologies on Z generated by sequences, extending results of Gabriyelyan [21]. More precisely, for a T-sequence b=(bn)n∈N of integers and the associated topology Tb on Z (in the sense of [28]), we prove that (Z,Tb) is reflexive whenever the ratios qn=bn+1bn are integers and diverge to ∞ (whereas the same conclusion was obtained in [21] under the more stringent condition ∑n≥11qn<∞). The character group of (Z,Tb) is the subgroup ttb(T):={x+Z∈T:bnx+Z→0} of the torus T. If the ratios qn are integers and for some ℓ∈N the sequence of quotients (bn+ℓbn) diverges to ∞, then ttb(T) with the compact-open topology is reflexive.

Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group ▵*(3,q,r) for q &lt; r Primes

Hyperbolic triangle groups are found within the category of finitely generated groups. These are topological groups formed by the reflections along the sides of a hyperbolic triangle and acting properly discontinuously on the hyperbolic plane. Higman raised a question about the simplicity of finitely generated groups. The best known example of a simple group is the alternating group An, where n≥5. This article establishes a relation between the hyperbolic triangle group denoted as ▵*(3,7,r) and the alternating group. The approach involves employing coset diagrams to establish this connection. The construction of adjacency matrices for these coset diagrams is performed, followed by a detailed examination of their spectral characteristics.

Representing topological full groups in Steinberg algebras and C*-algebras

We study the natural representation of the topological full group of an ample Hausdorff groupoid in the groupoid's complex Steinberg algebra and in its full and reduced C*-algebras. We characterise precisely when this representation is injective and show that it is rarely surjective. We then restrict our attention to discrete groupoids, which provide unexpected insight into the behaviour of the representation of the topological full group in the full and reduced groupoid C*-algebras. We show that the image of the representation is not dense in the full groupoid C*-algebra unless the groupoid is a group, and we provide an example showing that the image of the representation may still be dense in the reduced groupoid C*-algebra even when the groupoid is not a group.