# Questions tagged [amenability]

The amenability tag has no usage guidance.

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### 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 ...

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### 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 ...

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114 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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### 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 ...

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### 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 ...

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258 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 ...

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### 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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### 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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### 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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### 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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### 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 ...

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118 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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### 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 ...

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### 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 ...

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### 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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### 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 ...

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263 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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### 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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### 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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### 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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253 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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### 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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### 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 $(G,\tau_1)$ ...

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $G$).
...

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### 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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### 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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### 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 ...

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### Averaging measurable functions over amenable group actions

Let $G$ be an amenable group acting on a space $X$.
Amenability means there is a $G$-invariant mean on $L^\infty(G,{\mathbf R})$.
Given a bounded function $f\colon X\to {\mathbf R}$ one can use the ...

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### Semi-simple Banach algebra

Is there an example of an unital commutative semi-simple Banach algebra which it is not amenable?

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### Infinite topological direct sum of amenable Banach algebra

Is an infinite topological direct sum of amenable Banach algebras amenable again?
Can you give me a good reference about this notion?
Thanks

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### Free actions of non-amenable groups

Let $G$ be a locally compact, second countable, non-amenable group, let $X$ be a Haudorff space that is not necessarily compact, and let $G \curvearrowright X$ be a topological action that is free (i....

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### Non-ergodic Dye Theorem for orbit equivalent automorphisms

The Dye Theorem states that any two free ergodic p.m.p automorphisms of a standard probability space are orbit-equivalent.
Question: Is there a version of the above theorem for non-ergodic ...

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### Can the Cesaro limit of a positive definite function be negative?

Let $G$ be a countable amenable group and $\gamma:G\to\mathbb{C}$ a positive (semi)definite function (i.e. such that $\gamma(g^{-1})=\overline{\gamma(g)}$ and
$$\sum_{g,h\in G}f(g)\overline{f(h)}\...

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### Does the Approximation Property (AP) pass to quotients by amenable subgroups?

Given a countable group $G$ with the AP, and an normal, amenable subgroup $N$ of $G$, does $G/N$ have the AP?
In particular, does there exist a group $G$ with the AP and a surjective group ...

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### Can the full and reduced group $C^*$-algebras be “noncanonically” isomorphic?

Is there a locally compact group $G$ such that the canonical map from $C^{*}(G)$ to $C^{*}_{red} G$ is not isomorphism, hence $G$ is not amenable but these two $C^{*}$ algebras are isomorphic ...

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### Characterizing residually amenable groups

Let $G$ be a finitely generated group. The amenability of $G$ is equivalent to the existence of a certain "weak measure" on $G$. Is there such a characterization for residually amenable groups as well?...

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### What makes the amenability of Thompsons group $F$ such a tricky problem?

An open problem that seems to get a lot of attention every once in a while is the amenability of Thompsons group $F$.
The problem seems to generate both proofs and disproofs at a fairly high rate, ...

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### Actions of amenable groups on graphs with uncountably many ends

Let $G$ be a finitely generated amenable group acting transitively on an amenable Schreier graph $S$. Is it possible for $S$ to have uncountably many ends? An amenable graph with uncountably many ends ...