# 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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### Introductory text on amenability

I am looking for a book that covers amenability rigorously. Preferably a book aimed at beginners.
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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
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 ...