Questions tagged [amenability]

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Spectral radius of a finitely generated group

Let $G$ be a finitely generated group and $\Gamma$ be its Cayley graph with the usual word metric. Let $\mu$ be a symmetric non-degenerate measure on $G$ (maybe with finite support or smooth), and ...
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4 votes
0 answers
97 views

Nuclear approximations preserving the Cartan sub-algebra structure

Let $D \subset A$ be a unital Cartan inclusion, that is, $A$ is a separable $C^*$-algebra, and $D$ is a maximally abelian sub-$C^*$-algebra that contains the identity of $A$, with the property that $A$...
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10 votes
0 answers
183 views

Minimal actions commuting with amenable actions of $\mathbb{F}_2$

For a countable discrete group $G$ acting by homeomorphisms on a compact metrizable space $X$, we say that $G\curvearrowright X$ is (topologically) amenable if there exists a sequence of continuous ...
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1 vote
0 answers
65 views

Closed subgroups of totally disconnected Polish amenable groups

Let $G$ be a totally disconnected Polish topological group (e.g., a closed subgroup of the homeomorphism group of the Cantor set). If $G$ is amenable, is every closed subgroup of $G$ also amenable? ...
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2 votes
0 answers
127 views

Closure of an amenable subgroup

Let $G$ be a topological group, and let $H < G$ be a countable subgroup that is amenable as a discrete group. Is the closure of $H$ an amenable topological group?
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3 votes
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Amenability, growth and asymptotic dimension

I recently found this question on MSE, relating growth of groups with whether they are amenable, elementary amenable or not. I would like to know if there is an extra relation to finite or infinite ...
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9 votes
1 answer
182 views

Do extensions of pure states separate points?

Let $B$ be a unital C*-algebra and let $A⊆B$ be a closed *-subalgebra containing the unit of $B$. I am mostly interested in the case that $A$ is abelian but, for the strict purpose of stating my ...
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6 votes
1 answer
270 views

Trans-amenability of group actions

This problem is derived from this post. Let $G$ be a countable discrete group and $H\le G$ be a subgroup. Consider the $G$-action on $X=G/H$. Then the following amenability-like conditions are ...
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8 votes
0 answers
134 views

Uniform amenability at infinity

Let's recall that a group $G$ is amenable if for any finite subset $E\subset G$ and any $\epsilon>0$ there is a finite subset $F\subset G$ such that $$\max_{s\in E} |s F \mathbin{\triangle} F| \le \...
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7 votes
2 answers
590 views

Amenable action intuition

Let $\Gamma$ be a discrete group and $A$ be a $C^*$-algebra. Consider an action $\alpha: \Gamma \to \operatorname{Aut}(A)$. There is a notion of amenability for such an action (see e.g. Brown and ...
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38 votes
1 answer
3k views

Is there any version of the Banach-Tarski paradox in ZF?

The Banach-Tarski paradox states that for a solid ball in 3‑dimensional space, there exists a decomposition into a finite number of disjoint subsets, which can then be put back together in a different ...
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16 votes
3 answers
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What are the main open problems in the theory of amenability of groups?

I have read the Paterson and Runde books about amenability of groups, but I do not know what are the most intriguing questions in this area today. A survey or a list of questions would be welcome.
8 votes
0 answers
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Amenable automatic groups

Are there any known examples of finitely generated groups that are both amenable and automatic, besides the easy example of virtually abelian groups? Or are there any known restrictions that arise if ...
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7 votes
0 answers
117 views

Can a lacunary hyperbolic group be Liouville?

While discussing with a colleague of a possible use of lacunary hyperbolic elementarily amenable groups introduced by Olshanskii, Osin & Sapir, it occurred to me that I was not aware whether ...
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2 votes
1 answer
90 views

Is it possible to characterize the elements of the C$^*$-algebra of an open subgroupoid?

$\newcommand{\Cstar}{C^*_{\text{red}}}\newcommand{\G}{\mathscr G}\newcommand{\H}{\mathscr H}$Let $\G$ be an etale groupoid, let $U$ be an open subset of $\G^{(0)}$, and let $$ \H = \{\gamma \in \G:...
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4 votes
0 answers
142 views

Is the group ring of an amenable group, viewed as multiplicative monoid, amenable?

Motivated by this question, it seems natural to ask the following: Question 1: Is there a [finitely generated discrete] torsion-free virtually Abelian (but not Abelian) group $G$ so that the ...
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6 votes
1 answer
234 views

Amenable groupoid C*-algebras satisfy the UCT in English?

As is by now well known, Tu proved in 1998 that the C*-algebras coming from amenable groupoids satisfy the so-called UCT (universal coefficient theorem). Unfortunately, I don't speak french and I've ...
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3 votes
3 answers
313 views

Følner sequences with weird shapes

Let $G$ be a discrete and finitely generated group. Recall that $\{F_n\}_{n \in \mathbb{N}}$ is a Følner sequence if $|g F_n \cup F_n|/|F_n| \rightarrow 1$ for every $g \in G$. As is well known, ...
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0 votes
1 answer
67 views

Is there a $G$-paradoxical $G$-invariant subset of the plane for $G$ a group of rigid motions?

The Sierpinski-Mazurkiewicz paradox yields a nonempty rigid-motion paradoxical subset $S$ of the Euclidean plane: $S$ is the disjoint union of $A$ and $B$, each of which is $G$-equidecomposable with $...
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11 votes
1 answer
289 views

Does every topological group embed as a closed subgroup in an amenable group?

It is a standard result that closed subgroups of locally compact amenable groups are themselves amenable, so for example $F_2$, the free group on two generators, cannot be embedded as a closed ...
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0 votes
2 answers
160 views

Induced probability measure on a finite orbit under a group action

Suppose we have a discrete group $G$ acting on a compact set $X \subseteq \mathbb{R}^d$ via measure-preserving homeomorphisms, and suppose we have a point $x$ whose orbit $Gx$ is finite (say $|Gx| = n$...
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4 votes
2 answers
556 views

Is the set of all ICC amenable groups countable?

Is the set of all ICC amenable groups countable? If "yes", then in general, the classes of all countable ICC groups that give rise to the same von Neumann algebra (factor) -- are these ...
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1 vote
1 answer
55 views

Invariant strictly positive hyperreal probability measures on groups

Under what conditions is there a strictly positive hyperreal probability measure on a group $G$? This would be a finitely-additive non-negative function $\mu$ from the powerset of $G$ to a hyperreal ...
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1 vote
1 answer
127 views

What is the relation between the Tarski number and growth of a group?

When I was studying the structure of the Grigorchuk group, a question came to my mind and I just had the following information: We know that every finitely generated group of subexponential growth is ...
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2 votes
0 answers
129 views

About Tarski number 7

Recall that a group $G$ admits a $(m+n)$-paradoxical decomposition if there exist positive integers $m$ and $n$, a partition $\{P_1,\dotsc,P_m,Q_1,\dotsc,Q_n\}$ of $G$ and elements $x_i, y_j$ of $G$ ...
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4 votes
1 answer
220 views

Latest progress on Tarski numbers

Two questions, the first: What is the smallest non negative integer that we do not know yet is the Tarski number of a group? The second question is the same as in the title: What is the latest ...
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5 votes
2 answers
297 views

Examples of amenable Banach algebras which have non-amenable subalgebra

I am looking for examples of amenable Banach algebras which have non-amenable subalgebra I know 1: Each amenable Banach algebra has a bounded approximate identity 2: If $I$ be a closed ideal in an ...
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2 votes
2 answers
147 views

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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6 votes
1 answer
197 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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2 votes
0 answers
57 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 ...
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8 votes
2 answers
398 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 ...
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2 votes
1 answer
350 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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1 vote
1 answer
150 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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4 votes
1 answer
349 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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2 votes
0 answers
43 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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4 votes
0 answers
126 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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0 votes
0 answers
38 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 ...
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  • 763
12 votes
1 answer
326 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 ...
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5 votes
1 answer
222 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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3 votes
0 answers
152 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 ...
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4 votes
3 answers
383 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 ...
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  • 6,053
4 votes
1 answer
272 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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3 votes
1 answer
224 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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12 votes
1 answer
545 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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0 votes
0 answers
66 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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18 votes
2 answers
1k 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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  • 515
3 votes
0 answers
214 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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5 votes
1 answer
268 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 ...
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4 votes
1 answer
136 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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10 votes
1 answer
202 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 ...
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