# Amenability of Closed Subgroups and Orlicz Spaces

@article{Kopylov2013AmenabilityOC, title={Amenability of Closed Subgroups and Orlicz Spaces}, author={Yaroslav Kopylov}, journal={arXiv: Representation Theory}, year={2013} }

We prove that a closed subgroup $H$ of a second countable locally compact group $G$ is amenable if and only if its left regular representation on an Orlicz space $L^\Phi(G)$ for some $\Delta_2$-regular $N$-function $\Phi$ almost has invariant vectors. We also show that a noncompact second countable locally compact group $G$ is amenable if and ony if the first cohomology space $H^1(G,L^\Phi(G))$ is non-Hausdorff for some $\Delta_2$-regular $N$-function $\Phi$.

## 9 Citations

### $\Phi$-Harmonic Functions on Discrete Groups and First $\ell^\Phi$-Cohomology

- Mathematics
- 2013

We study the first cohomology groups of a countable discrete group $G$ with coefficients in a $G$-module $\ell^\Phi(G)$, where $\Phi$ is an $N$-function of class $\Delta_2(0)\cap \nabla_2(0)$. In…

### Φ-Harmonic functions on discrete groups and the first ℓΦ-cohomology

- Mathematics
- 2014

We study the first cohomology groups of a countable discrete group G with coefficients in a G-module ℓΦ(G), where Φ is an N-function of class Δ2(0) ∩ ▿2(0). Developing the ideas of Puls and…

### Orlicz spaces and the large scale geometry of Heintze groups

- MathematicsMathematische Annalen
- 2016

We consider an Orlicz space based cohomology for metric (measured) spaces with bounded geometry. We prove the quasi-isometry invariance for a general Young function. In the hyperbolic case, we prove…

### Orlicz spaces and the large scale geometry of Heintze groups

- Mathematics
- 2014

We consider an Orlicz space based cohomology for metric (measured) spaces with bounded geometry. We prove the quasi-isometry invariance for a general Young function. In the hyperbolic case, we prove…

### Φ-Harmonic functions on discrete groups and the first ℓΦ-cohomology

- MathematicsSiberian Mathematical Journal
- 2014

We study the first cohomology groups of a countable discrete group G with coefficients in a G-module ℓΦ(G), where Φ is an N-function of class Δ2(0) ∩ ▿2(0). Developing the ideas of Puls and…

### The Rao–Reiter Criterion for the Amenability of Homogeneous Spaces

- MathematicsSiberian Mathematical Journal
- 2018

We prove that a homogeneous space G/H, with G a locally compact group and H a closed subgroup of G, is amenable in the sense of Eymard–Greenleaf if and only if the quasiregular action πΦ of G on the…

### The Rao–Reiter Criterion for the Amenability of Homogeneous Spaces

- MathematicsSiberian Mathematical Journal
- 2018

We prove that a homogeneous space G/H, with G a locally compact group and H a closed subgroup of G, is amenable in the sense of Eymard–Greenleaf if and only if the quasiregular action πΦ of G on the…

### On Asymptotic and Continuous Group Orlicz Cohomology

- Mathematics
- 2022

. We generalize some results on asymptotic and continuous group L p -cohomology to Orlicz cohomology. In particular, we show that asymptotic Orlicz cohomology is a quasi-isometry invariant and that…

### Orlicz Spaces and Amenability of Hypergroups

- MathematicsBulletin of the Iranian Mathematical Society
- 2019

In this note, we introduce and study some generalizations of Reiter’s condition (Pp)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}…

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