Questions tagged [amenability]

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Upper density of subsets of an amenable group

Let $G$ be an amenable group (so locally compact Hausdorff) and also assume it is second countable if needed. My question is that what are the standard ways (across literature) of defining the upper ...
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109 views

Introductory text on amenability

I am looking for a book that covers amenability rigorously. Preferably a book aimed at beginners.
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0answers
51 views

Measure invariant under circle maps

Consider continuous bijections (may even assume these are homeomorphisms or diffeomorphisms if it helps) from the circle onto itself given by $x \mapsto x + s_i(x)$ where $i = 1,2$ or $3$. (I'm ...
8
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2answers
348 views

Constant Martin kernel and amenability

Consider a finitely supported random walk on a discrete group G such that the support generates $G$ as a semigroup. The Martin kernels are then non-negative functions on the product $G \times M$ where ...
2
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1answer
264 views

Is a product of Følner sets Følner?

Let $G$ be an amenable (countable, discrete) group and let $F_1,F_2,...,F_n,...$ and $G_1,G_2,...,G_n,...$ be two Følner sequences. Is the product sequence (i.e. the sequence $(H_n)$ where $H_n$ is ...
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1answer
134 views

A question about Johnson's theorem on the first and second cohomology of commutative amenable algebras

Johnson in cohomology of Banach algebra proved the following proposition. I need to some guidance for the bold part of the following proof. Do you know any papers or book with more details for this ...
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1answer
655 views

How to rigorously define a translation-invariant measure that follows these requirements?

I am unsure anyone at math stack exchange can answer my question, so I moved it to MathOverflow. Let $A$ be a subset of $\mathbb{R}$. I want to rigorously define what I believe is the most "intuitive ...
4
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1answer
331 views

First and second cohomology groups of Banach algebras

Johnson in the introduction section (page 1) in "Cohomology in Banach algebras" ZBL0256.18014, wrote that Guichardet in [14,15] obtained for a Banach algebra $A$, one has $H^1(A,X)=H^2(A,X)=0$, ...
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41 views

Weak amenability hereditary properties

Let $\mathcal{A}$ be a commutative weakly amenable Banach algebra and $\mathcal{B}$ be a Banach algebra, let $\theta:\mathcal{A} \to \mathcal{B}$ be a continuous homomorphism with dense range; then it ...
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114 views

Contractible Banach algebras

A Banach algebra $A$ is contractible if $H^1(A, X)=0$ for all Banach $A$-bimodules $X$. Now to my question Let $A$ be Banach algebra and $I$ be closed ideal of $A$. If $I$ and $A/I$ are both ...
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37 views

If $P(G)=\{f\in L_1(G): f\geq 0, \int f d\lambda(x)=1\}$ Prove $\overline{\widehat{P(G)}}^*$ = $\mathfrak{M}(G)$

Let $\mathfrak{M}(G)$ be the set of all means on $L_\infty (G)$ If $P(G)=\{f\in L_1(G): f\geq 0, \int f d\lambda(x)=1\}$ Prove $\overline{\widehat{P(G)}}^* = \mathfrak{M}(G)$ My attempt: We know ...
9
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1answer
207 views

Topological amenability vs amenability of an action

Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space. Assume that $G$ acts on $X$ by homeomorphisms. Consider the following two definitions: [$C^*$-algebras and finite dimensional ...
4
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1answer
174 views

example of a non-amenable l.c. group such that $C_r^*(G)$ satisfies the UCT

Are there known any examples of non-amenable locally compact (or more restrictive, non-amenable discrete) groups $G$ for which the reduced group $C^*$-algebra $C_r^*(G)$ satisfies the universal ...
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138 views

Fixed-point properties for affine actions of topological groups

T. Mitchell [Illinois J. Math. 14 (1970) 630--641] defined four properties of a topological semigroup, and in particular of a topological group $G$. Two of them are: (F2) Every jointly continuous ...
2
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1answer
200 views

Amenable Thompson-like groups

Question: Do there exist amenable Thompson-like groups? I realise that my question is vague, but defining and studying groups which look like Thompson's groups $F$, $T$ and $V$ seems to be an ...
4
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1answer
262 views

A question about spectral properties of a non-amenable group

Let $G$ be a group generated by $a,b$ (for the sake of simplicity). Consider the element $$S=a+b+a^{-1}+b^{-1}\in{\mathbb C}[G],$$ which may also be interpreted as an operator in $l^2(G)$ (by left ...
3
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1answer
160 views

Is it possible to put Higman group as an amenable by sofic group?

I know Higman group has an amalgamated product decomposition of $BS(1, 2)$. Is it possible to decompose Higman group as some groups we are more familiar with. For example, is there a normal subgroup K ...
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1answer
490 views

Is there a notion of 'amenable ring'

Amenable groups are everywhere these days, as examples of all kinds of lovely phenomena. And there are various ways of defining notions of 'amenable monoid' or possibly 'amenable semigroup'. But for ...
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65 views

weakly amenable weighted sequence algebras

Let $v=(v_n)_{n\in\mathbb{N}}$ be a positive weight with $\inf_nv_n>0$ (for convenience we may take $v_n\geqslant1$). Then $\ell_{\infty}(v)$ is a Banach algebra with coordinate-wise multiplication....
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827 views

Can There be a 1 dimensional Banach-Tarski paradox in the absence of choice

Let $\mathbb{R}$ act on itself by translation. Then there is no finite decomposition of a unit interval into pieces which, when translated, yields two distinct unit intervals. More formally does ...
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158 views

Are there amenable groups without explicit Folner sets?

This is essentially a follow-up to this previous discussion on how, in the absence of choice, the "invariant mean" and "Folner set" characterizations of amenability are no longer equivalent. Recently ...
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1answer
229 views

Amenability of $S^{\infty}$

Let $G$ be the group of all permutations of $\mathbb{N}$. If I am not mistaken, this group is denoted by $S^{\infty}$. Is there a precise locally compact topology on $G$ such that $G$ would ...
4
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1answer
125 views

approximate diagonal

Let $I$ be an arbitary index set, $((A_i)_i,\|.\|_i)_{i\in I}$ be a family of Banach algebras, with approximate diagonal $(m^{i}_α)_α\subseteq A_i\hat\otimes A_i$, and $B=\{(x_i)_i\in {\displaystyle \...
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1answer
176 views

Naturally occurring, non-amenable Zappa-Szep products of discrete amenable groups?

We say $G$ is the Zappa-Szep product of two subgroups $K$ and $P$ if $K\cap P = \{e\}$ and the function $K\times P \to G$, $(k,p)\mapsto kp$, is bijective. The Iwasawa decomposition shows that we can ...
3
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1answer
86 views

About Beurling algebras

Do there exist an amenable Beurling algebra that is neither Arens regular nor strongly Arens irregular? In his memoir "The second duals of Beurling algebras", A. T. Lau proved that there exists a ...
6
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1answer
267 views

A group, neither amenable, nor having a subgroup that looks like $F_2$ up to level $n$?

It is known that there are non-amenable groups not containing $F_2$, the free group on two generators. We can even have that every 2-generated subgroup is finite. But is there a non-amenable group $...
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142 views

relative amenability of von Neumann algebra

Let $\cal{M}$ be a finite von Neumann algebra and $\cal{N}$ be a von Neumann subalgebra of $\cal{M}$. The von Neumann algebra $\cal{M}$ is is amenable relative to $\cal{N}$ if there exists a norm ...
9
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1answer
304 views

Does every non-amenable group contain a 2-generated non-amenable subgroup?

It is known that there are non-amenable groups not containing $F_2$, the free group on two generators; for example, Olshanskii's group. But does every non-amenable group contain a 2-generated non-...
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182 views

Superharmonic functions and amenability

Let $G$ be a group generated by a finite set $S$. Let $P$ be a Markov operator defined by the uniform measure on $S$. A function is superharmonic if $Pf\leq f$. Assume that there is a set of non-...
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1answer
96 views

Example of an amenable enveloping von-Neumann algebra

I am looking for an example of an infinite dimensional $C^{*}$-algebra whose second dual is amenable. Can anyone supply a suggested reference? Many thanks in advance. Edit: If this is ...
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1answer
164 views

Can an amenable group have a weak mixing unitary representation without almost invariant vectors?

Does there exist a finitely generated discrete amenable group $G$ that acts on a separable Hilbert space $\mathcal{H}$ by unitary transformations, and where (1) $\mathcal{H}$ has no finite dimensional ...
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2answers
346 views

Poisson Furstenberg Boundary of topological groups, reference request

I'm trying to understand the relations between between the following group properties, in the case of (say, compactly generated locally compact) topological groups: Group growth. Amenability. Poisson ...
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141 views

Are all invariant means on a group equally good?

Let $G$ be an infinite countable group. If $G$ is amenable then it has a number of other interesting properties. To prove such a property from the existence of an invariant mean on $G$, we usually ...
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133 views

amenable locally compact group

Let $\tau_1,\tau_2 $ be topologies on group $G$ such that $(G,\tau_1),(G,\tau_2)$ be a locally compact group. Let $\tau_1\subseteq\tau_2$ and $(G,\tau_2)$ be an amenable group, when $(G,\tau_1)$ ...
3
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1answer
192 views

Quotient of weak amenable Banach algebras

Let $A$ be a weak amenable Banach algebra and $I$ be a closed (two-sided) ideal of $A$. In general $\frac{A}{I}$ is not weakly amenable. Is there an example of this type of weak amenable Banach ...
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270 views

How to construct Følner sequences?

Q1. Are there "canonical" ways to construct Følner sequences for locally compact amenable groups? Possibly via representations or perhaps characters ? To elaborate, let $G$ be a l.c. amenable group ...
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154 views

Existence of any strong form of Folner condition for elementary amenable groups

Recall that a discrete group $G$ is amenable if and only if it has Folner condition, i.e., for every positive number $\epsilon$ and every finite set $A$ of $G$ there exists a finite non-empty subset $...
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1answer
200 views

Kernels of representations of $C^*(G)$

Let G be a discrete group. I am interested in the following: let $\pi$ and $\rho$ be two representations of $G$. Denote by $C^*Ker\pi$ and $C^* Ker \rho$ the kernels of the corresponding ...
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88 views

Amenability for Actions twited with 2-cocycles

Let $A \subset B(H)$ be a unital $C^\ast$-algebra and $\theta: G \rightarrow \mathrm{Aut}(A)$ an action and let $\omega: G \times G \rightarrow U(\mathcal{Z}(A))$ be a $2$-cocyle with respect to $\...
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182 views

Inner amenability

There is a well-known result of Rosenblatt that if $G$ is non-amenable discrete group acting on the space $X$ and a stabilizer of each point $x\in X$ is amenable then the action of $G$ on itself is ...
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270 views

The Burnside group B(2,5) and amenability

Let $n \ge 2$, then the Burnside group $B(n,m) = \langle a_1, \dots , a_n \mid \forall \omega, \ \omega^m = e \rangle$ is known to be non-amenable for odd $m \ge 665$, and finite for $m = 2,3,4,6$, ...
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1answer
244 views

Amenable inverse limits of torsionfree amenable groups

Let $$ \cdots \to \Gamma_n \to \Gamma_{n-1} \to \cdots \to \Gamma_0$$ be an inverse system countable groups and let's assume (for this post) that all homomorphisms in such an inverse system are ...
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264 views

Is there an countable amenable dense subgroup of $U(\ell^2 \mathbb N)$?

Question: Does the unitary group $U(\ell^2 \mathbb N)$, equipped with the strong operator topology, contain a countable dense subgroup which is amenable as a discrete group? I would be also ...
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257 views

Inner amenable groups with property (T)

It is well-known that amenable groups which have property T are necessarily compact. I am interested if the situation is the same for inner amenable group with property T? Especially if the group is ...
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342 views

Elementary classes of groups

For a given class of groups, called base class, $B$. Osin have defined elementary closure of $B$ as the minimal class that contains $B$ and is closed under taking subgroups, extensions, direct unions, ...
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173 views

Johnson's Theorem - Proof (Runde) Clarification

I am reading Runde's Lectures on Amenability. In the proof of Johnson's theorem where he proves "$L^{1}(G)$ is amenable Banach algebra implies $G$ is amenable" : he defines a $L^1(G)$ bimodule action ...
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Finitely presented amenable LERF group which is not virtually solvable

Is there a group $G$ with the following properties? Finitely presented Amenable Not virtually solvable LERF (that is, every finitely generated subgroup is closed in the profinite topology on $G$). ...
7
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2answers
545 views

Thompson group $V$

R. Thompson introduced three groups $F\subset T\subset V$. The question concerning amenability of $F$ is still unanswered and has attracted much attention. I have read that Thompson group $V$ contains ...
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2answers
328 views

Are finitely generated amenable groups positively finitely generated?

Let $G$ be a finitely generated amenable group. Is there a positive integer $n$ such that $n$ random elements of $G$ generate it with positive probability? Being more formal, note that $G^n$ is ...
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75 views

Almost invariance in compact quotients of locally compact groups

While trying to get an analogue of Weiss's monotiling result for amenable residually finite groups in the topological setting, I face the following problem. Let $G$ be a locally compact amenable ...