Abstract measure algebras over homogeneous spaces of compact groups

2018 ◽  
Vol 29 (01) ◽  
pp. 1850005 ◽  
Author(s):  
Arash Ghaani Farashahi
Keyword(s):  
Compact Group ◽  
Systematic Study ◽  
Measure Space ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Compact Groups ◽  
Coset Space ◽  
Left Coset ◽  
Abstract Measure

This paper presents a systematic study for abstract Banach measure algebras over homogeneous spaces of compact groups. Let [Formula: see text] be a closed subgroup of a compact group [Formula: see text] and [Formula: see text] be the left coset space associated to the subgroup [Formula: see text] in [Formula: see text]. Also, let [Formula: see text] be the Banach measure space consists of all complex measures over [Formula: see text]. Then we introduce the abstract notions of convolution and involution over the Banach measure space [Formula: see text].

Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups

2017 ◽  
Vol 60 (1) ◽  
pp. 111-121 ◽  
Author(s):  
Arash Ghaani Farashahi
Keyword(s):  
Hilbert Space ◽  
Operator Theory ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Compact Groups ◽  
Theory Approach ◽  
Coset Space ◽  
Trace Formulas ◽  
Left Coset ◽  
Abstract Notion

AbstractThis paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. LetGbe a compact group and letHbe a closed subgroup ofG. LetG/Hbe the left coset space ofHinGand letμbe the normalized G-invariant measure onG-Hassociated with Weil’s formula. Then we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert spaceL2(G/H,μ).

A Class of Abstract Linear Representations for Convolution Function Algebras over Homogeneous Spaces of Compact Groups

2018 ◽  
Vol 70 (1) ◽  
pp. 97-116 ◽  
Author(s):  
Arash Ghaani Farashahi
Keyword(s):  
Invariant Measure ◽  
Compact Group ◽  
Homogeneous Space ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Compact Groups ◽  
Compact Homogeneous Space ◽  
Linear Representations ◽  
Function Algebras ◽  
Convolution Function

AbstractThis paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. LetGbe a compact group andHa closed subgroup ofG. Letμbe the normalizedG-invariant measure over the compact homogeneous spaceG/Hassociated with Weil's formula and. We then present a structured class of abstract linear representations of the Banach convolution function algebrasLp(G/H,μ).

ABSTRACT HARMONIC ANALYSIS OF RELATIVE CONVOLUTIONS OVER CANONICAL HOMOGENEOUS SPACES OF SEMIDIRECT PRODUCT GROUPS

2016 ◽  
Vol 101 (2) ◽  
pp. 171-187 ◽  
Author(s):  
ARASH GHAANI FARASHAHI
Keyword(s):  
Harmonic Analysis ◽  
Homogeneous Space ◽  
Semidirect Product ◽  
Homogeneous Spaces ◽  
Continuous Homomorphism ◽  
Compact Groups ◽  
Coset Space ◽  
Unified Approach ◽  
Left Coset ◽  
Semidirect Product Groups

This paper presents a structured study for abstract harmonic analysis of relative convolutions over canonical homogeneous spaces of semidirect product groups. Let $H,K$ be locally compact groups and $\unicode[STIX]{x1D703}:H\rightarrow \text{Aut}(K)$ be a continuous homomorphism. Let $G_{\unicode[STIX]{x1D703}}=H\ltimes _{\unicode[STIX]{x1D703}}K$ be the semidirect product of $H$ and $K$ with respect to $\unicode[STIX]{x1D703}$ and $G_{\unicode[STIX]{x1D703}}/H$ be the canonical homogeneous space (left coset space) of $G_{\unicode[STIX]{x1D703}}/H$. We present a unified approach to the harmonic analysis of relative convolutions over the canonical homogeneous space $G_{\unicode[STIX]{x1D703}}/H$.

scholarly journals Recurrence and ergodicity of random walks on linear groups and on homogeneous spaces

2011 ◽  
Vol 32 (4) ◽  
pp. 1313-1349 ◽  
Author(s):  
Y. GUIVARC’H ◽  
C. R. E. RAJA
Keyword(s):  
Random Walks ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Local Fields ◽  
Quadratic Growth ◽  
Linear Groups ◽  
Compact Groups ◽  
Large Classes ◽  
Skew Products ◽  
Recurrent Random Walk

AbstractWe discuss recurrence and ergodicity properties of random walks and associated skew products for large classes of locally compact groups and homogeneous spaces. In particular, we show that a closed subgroup of a product of finitely many linear groups over local fields supports an adapted recurrent random walk if and only if it has at most quadratic growth. We give also a detailed analysis of ergodicity properties for special classes of random walks on homogeneous spaces and for associated homeomorphisms with infinite invariant measure. The structural properties of closed subgroups of linear groups over local fields and the properties of group actions with respect to certain Radon measures associated with random walks play an important role in the proofs.

scholarly journals Г-invariant operators associated with locally compact groups

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2721-2730
Author(s):  
Ali Ghaffari ◽  
Somayeh Amirjan
Keyword(s):  
Compact Group ◽  
Closed Subgroup ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Usual Sense ◽  
Left Translation ◽  
Compact Groups ◽  
Locally Compact ◽  
Product Group

Let G be a locally compact group and let ? be a closed subgroup of G x G. In this paper, the concept of commutativity with respect to a closed subgroup of a product group, which is a generalization of multipliers under the usual sense, is introduced. As a consequence, we obtain characterization of operators on L2(G) which commute with left translation when G is amenable.

Finitely generated subgroups and Chabauty topology in totally disconnected locally compact groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hatem Hamrouni ◽  
Yousra Kammoun
Keyword(s):  
Positive Integer ◽  
Compact Group ◽  
Closed Subgroup ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Finitely Generated ◽  
Locally Compact ◽  
Chabauty Topology ◽  
Totally Disconnected

Abstract For a locally compact group 𝐺, we write S ⁢ U ⁢ B ⁢ ( G ) {\mathcal{SUB}}(G) for the space of closed subgroups of 𝐺 endowed with the Chabauty topology. For any positive integer 𝑛, we associate to 𝐺 the function δ G , n \delta_{G,n} from G n G^{n} to S ⁢ U ⁢ B ⁢ ( G ) {\mathcal{SUB}}(G) defined by δ G , n ⁢ ( g 1 , … , g n ) = gp ¯ ⁢ ( g 1 , … , g n ) , \delta_{G,n}(g_{1},\ldots,g_{n})=\overline{\mathrm{gp}}(g_{1},\ldots,g_{n}), where gp ¯ ⁢ ( g 1 , … , g n ) \overline{\mathrm{gp}}(g_{1},\ldots,g_{n}) denotes the closed subgroup topologically generated by g 1 , … , g n g_{1},\ldots,g_{n} . It would be interesting to know for which groups 𝐺 the function δ G , n \delta_{G,n} is continuous for every 𝑛. Let [ HW ] [\mathtt{HW}] be the class of such groups. Some interesting properties of the class [ HW ] [\mathtt{HW}] are established. In particular, we prove that [ HW ] [\mathtt{HW}] is properly included in the class of totally disconnected locally compact groups. The class of totally disconnected locally compact locally pronilpotent groups is included in [ HW ] [\mathtt{HW}] . Also, we give an example of a solvable totally disconnected locally compact group not contained in [ HW ] [\mathtt{HW}] .

scholarly journals Invariant Measures on Coset Spaces

1961 ◽  
Vol 5 (2) ◽  
pp. 80-85 ◽  
Author(s):  
S. Świerczkowski
Keyword(s):  
Closed Subgroup ◽  
Invariant Measures ◽  
Locally Compact Group ◽  
Coset Space ◽  
Modular Functions ◽  
Natural Topology ◽  
Left Coset ◽  
Generic Element ◽  
Baire Measure ◽  
Coset Spaces

In this note we consider measures on a left coset space G/H, where G is a locally compact group and H is a closed subgroup. We assume the natural topology in G/H and we denote the generic element of this space by xH (x∈G). Every element t∈G defines a homeomorphism of G/H given by t(xH) = (tx)H. A. Weil showed that a Baire measure on G/H invariant under all these homeomorphisms can exist only ifΔ(ξ) = δ(ξ) for each ξ ∈ H,where Δ(x), δ(ξ) denote the modular functions in G, H [6, pp. 42–45]. We shall devote our investigations to inherited measures on G/H (cf. [3] and the definition below) invariant under homeomorphisms belonging to a normal and closed subgroup T ⊂ G.

scholarly journals Invariant measures on double coset spaces

1965 ◽  
Vol 5 (4) ◽  
pp. 495-505 ◽  
Author(s):  
Teng-Sun Liu
Keyword(s):  
Compact Group ◽  
Closed Subgroup ◽  
Invariant Measures ◽  
Double Coset ◽  
Locally Compact Group ◽  
Coset Space ◽  
Canonical Mapping ◽  
Locally Compact ◽  
Coset Spaces ◽  
Image Position

Let G be a locally compact group with left invariant Haar measure m. Le H be a closed subgroup of G and K a compact group of G. Let R be the equivalence relation in G defined by (a, b)∈R if and if a = kbh for some k in K and h in H. We call E =G/R the double coset space of G modulo K and H. Donote by a the canonical mapping of G onto E. It can be shown that E is a locally compact space and α is continous and open Let N be the normalizer of K in G, i. e. .

scholarly journals Compact groups with a set of positive Haar measure satisfying a nilpotent law

2021 ◽  
pp. 1-4
Author(s):  
ALIREZA ABDOLLAHI ◽  
MEISAM SOLEIMANI MALEKAN
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Positive Answer ◽  
Nilpotent Subgroup ◽  
Compact Groups

Abstract The following question is proposed by Martino, Tointon, Valiunas and Ventura in [4, question 1·20]: Let G be a compact group, and suppose that \[\mathcal{N}_k(G) = \{(x_1,\dots,x_{k+1}) \in G^{k+1} \;|\; [x_1,\dots, x_{k+1}] = 1\}\] has positive Haar measure in $G^{k+1}$ . Does G have an open k-step nilpotent subgroup? We give a positive answer for $k = 2$ .

scholarly journals Ends of locally compact groups and their coset spaces

1974 ◽  
Vol 17 (3) ◽  
pp. 274-284 ◽  
Author(s):  
C. H. Houghton
Keyword(s):  
Compact Group ◽  
Direct Product ◽  
Compact Space ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Countable Base ◽  
Locally Compact ◽  
Coset Spaces ◽  
Compact Boundary

Freudenthal [5, 7] defined a compactification of a rim-compact space, that is, a space having a base of open sets with compact boundary. The additional points are called ends and Freudenthal showed that a connected locally compact non-compact group having a countable base has one or two ends. Later, Freudenthal [8], Zippin [16], and Iwasawa [11] showed that a connected locally compact group has two ends if and only if it is the direct product of a compact group and the reals.

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