The amenability tag has no wiki summary.

**5**

votes

**0**answers

230 views

### 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 ...

**3**

votes

**1**answer

181 views

### 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 ...

**4**

votes

**1**answer

214 views

### 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 ...

**16**

votes

**1**answer

750 views

### 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, ...

**4**

votes

**0**answers

86 views

### 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 ...

**4**

votes

**1**answer

204 views

### When does convolution preserve the `size' of a function?

For a positive function $f$ and positive measures $\mu, \nu$, does
$$\mu\ast f\leq \nu\ast f \Rightarrow \|\mu\|\leq \|\nu\|?$$
More details: Let $G$ be a locally compact group, $C(G)$ be the space ...

**1**

vote

**1**answer

159 views

### Infinite amenable group subfactors

Let amenable groups $\Gamma$ and $\Gamma'$. They act outerly of only one manner on the hyperfinite ${\rm II}_1$-factor $\mathcal{R}$.
Question: $(\mathcal{R} \subset \mathcal{R} \rtimes \Gamma) ...

**2**

votes

**1**answer

185 views

### ${\rm II}_1$-factors with finite commutant: $\mathcal{A} \cap \mathcal{B} = \mathbb{C} \Rightarrow \mathcal{A}' \cap \mathcal{B}'$ hyperfinite?

Let $\mathcal{A} , \mathcal{B} \subset B(H)$ be ${\rm II}_1$-factors such that $\mathcal{A}', \mathcal{B}' $ are also a ${\rm II}_1$-factors.
Question: $\mathcal{A} \cap \mathcal{B} = ...

**11**

votes

**0**answers

192 views

### Do quotients of amenable groups C*-algebras satisfy the UCT?

Let G be a discrete amenable group.
General Question: Let $J$ be an ideal of $C^*(G)$, the group C*-algebra of $G.$ Does $C^*(G)/J$ satisfy the universal
coefficient theorem (UCT)?
I am mainly ...

**9**

votes

**1**answer

198 views

### Without AC, which implications between the different definitions of amenability still hold?

More precisely, I would like to know which implications between the following definitions of amenability of a discrete countable (or even finitely generated) group can be proved to hold with only ZF ...

**3**

votes

**1**answer

164 views

### Is it compatible with ZF to assume that every amenable discrete group is finite?

The question is in the title, amenability being understood as the existence of a left-invariant finitely additive probability measure on the group of interest. The case of countable groups is treated ...

**5**

votes

**0**answers

185 views

### Is translation by the free group (in two generators) on a certain completion of the group an amenable action?

Let $\mathbb{F}_2 = \langle a,b\rangle$ be the free group in two generators $a,b$ and let $\alpha \in \text{End}(\mathbb{F}_2)$ be given by $\alpha(a) = a^2, \alpha(b)= b^2$. Note that the index ...

**0**

votes

**1**answer

121 views

### Amenability of the Koopman representation

Let $G$ be a locally compact group which acts non-singularly on a standard probability space $(X,\mu)$. Consider the Koopman representation $\pi_X:G\rightarrow U(L^2(X,\mu))$ defined by
...

**8**

votes

**0**answers

294 views

### Finitely additive measures on $\mathbb Z_2^\omega$ with invariance and independence constraints

Let $G = \mathbb Z_2^\omega$, with pointwise addition. Assume the Axiom of Choice. I am interested in finitely additive probability measures $\mu$ defined on all of $\mathcal PG$ that can be ...

**1**

vote

**2**answers

269 views

### Amenability of $l^\infty$ [closed]

I'm working on the amenability of some Banach algebras, and I'm wondering why $l^\infty$ is amenable ? Does any one has any idea how to start ?
Thank you in advance.

**3**

votes

**0**answers

107 views

### Pointwise ergodic theorem for amenable semigroups

Using tempered Folner sequences one may show a pointwise ergodic theorem for amenable groups.
(see http://www.aimsciences.org/journals/pdfsnews.jsp?paperID=2413&mode=full)
Is there a similar ...

**6**

votes

**1**answer

166 views

### Do syndetic sets on amenable semigroups have positive upper density?

Let $\mathbb{G}$ be a discrete amenable semigroup, and $\left\{ F_{n}\right\} $
a Folner sequence.
For $S\subset \mathbb{G}$ define the upper density as $D^{\ast
}(S)=\limsup_{n\rightarrow \infty ...

**2**

votes

**1**answer

192 views

### Two kinds of invariance of full conditional probabilities

Given a field $F$ of subsets of $\Omega$, we can define full conditional probabilities to be a function $P:F\times (F-\{ \varnothing \}) \to [0,1]$ such that:
$P(-|B)$ is a finitely-additive ...

**4**

votes

**0**answers

69 views

### Estimates for simple random walks in groups of intermediate growth

I'm looking for references for the rate of escape and return probability for a group of intermediate growth.
Let $0<\alpha < 1$. If the volume growth is $\succeq \mathrm{exp}(n^\alpha)$, then ...

**2**

votes

**1**answer

158 views

### Relative amenability of subgroups

Let $\Gamma$ be a countable group and let $\Lambda_1,\Lambda_2<\Gamma$ be subgroups. We say that $\Lambda_1$ is amenable relative to $\Lambda_2$ if the action of $\Lambda_1$ on $\Gamma/\Lambda_2$ ...

**10**

votes

**0**answers

135 views

### Star-shaped Folner sequence

Fix a (finite) generating set $S$ for $\Gamma$ (discrete) amenable. Given a Følner sequence (i.e. a sequence of finite sets $F_n$ whose boundary $\partial F_n$ in the Cayley graph of $S$ is such that ...

**0**

votes

**0**answers

101 views

### Amenability of the pair $(GL(2,\mathbb{Q})^+,SL(2,\mathbb{Z}))$

I am trying to see whether the pair $(GL(2,\mathbb{Q})^+,SL(2,\mathbb{Z}))$ is amenable in the following sense:
Let $H$ be a closed subgroup of a locally compact group $G$. The pair $(G,H)$ is called ...

**9**

votes

**0**answers

172 views

### Proving amenability of an extension by using paradoxical decompositions

It is well known that an extension of an amenable group by an amenable group is amenable. Is it possible to prove that by using only paradoxical decompositions: if $G$ has a paradoxical decomposition ...

**2**

votes

**1**answer

194 views

### Does supramenability imply that $a+c=b+2c \Rightarrow a=b+c$ on the type semigroup?

Tarski proved that if a group $G$ is exponentially bounded, then for $a$, $b$ and $c$ in the associated (equidecomposability) type semigroup, we have $a+c=b+2c \Rightarrow a=b+c$.
Question: Can ...

**3**

votes

**1**answer

198 views

### A stronger version of supramenability?

A group $G$ is supramenable iff for all $\varnothing\ne A\subseteq G$ there is a finitely-additive left-$G$-invariant measure $\mu_A$ on $G$ with $\mu_A(A)=1$. I'm interested in a seemingly stronger ...

**8**

votes

**0**answers

344 views

### The approximation property of group C*-algebras

Let $G$ be a discrete group. Then the group C*-algebra $C^*(G)$ is nuclear if and only if $G$ is amenable. I am wondering whether nuclearity of $C^*(G)$ can fail for a Banach-space reason, namely due ...

**55**

votes

**1**answer

3k views

### Non-amenable groups with arbitrarily large Tarski number?

Just out of curiosity, I wonder whether there are non-amenable groups with arbitrarily large Tarski numbers. The Tarski number $\tau(G)$ of a discrete group $G$ is the smallest $n$ such that $G$ ...

**1**

vote

**1**answer

112 views

### Amenable normal closure

Prove or disprove:
Let $G$ be a countable group. Let $H < G$ be an amenable subgroup with a finite conjugacy class. Then the normal closure of $H$ is also amenable.
Thanks!

**6**

votes

**1**answer

679 views

### Characterization of amenable actions

Let $(X,\mu)$ be a $G$-space, i.e. a measure space with a Borel quasi-invariant $G$-action. Say that $X$ is amenable (equivalently, that the action is amenable) if there is a $G$-fixed point in every ...

**4**

votes

**1**answer

337 views

### Probabilities of a random walk exiting a set

Let $F$ be a finite connected set in a graph (soon to be the Cayley graph of a group) and $\mathrm{Ex}_x^F$ be the function on the vertices in $F^c$ which are neighbour to vertices in $F$ defined as ...

**3**

votes

**0**answers

96 views

### Cogrowth and value of its series at the critical exponent

Let $G$ be a finitely generated group and write $G = F/N$ for $N$ a normal subgroup of a free group $F$. Let $S_n$ be the elements in $F$ written as words of exactly $n$ letters. So, for $n\geq 1$, ...

**3**

votes

**1**answer

181 views

### Spectral synthesis for central functions on locally compact groups

There is a large literature on harmonic analysis on locally compact group, that
I am just beginning to discover. However I have not seen so far anything that emphasizes the central functions on $G$. A ...

**16**

votes

**3**answers

441 views

### Is the isomorphism problem for amenable groups decidable?

Is it algorithmically decidable if two finitely presented amenable groups are isomorphic?
Or slightly different:
Does there exist a family of amenable groups (indexed by natural numbers) for which ...

**3**

votes

**1**answer

386 views

### Existence of nice Folner sequences

I'm attempting a proof by induction and, for the inductive step, it would be very useful for me to have some control on a Folner sequence. Indeed, let $G$ be a finitely generated amenable group, fix a ...

**5**

votes

**1**answer

213 views

### Cocycles for right- and left- regular representations on $\ell_2(G)$

Using the standard notation, let $\lambda_gf(x)=f(g^{-1}x)$ be the left regular representation and $\rho_gf(x)=f(xg)$ be the right regular one, acting on the space $V$ of complex-values functions on ...

**3**

votes

**2**answers

516 views

### Another question about amenability and Følner sequences

Følner's characterization of Amenability says that a group $G$ is amenable if there exists a directed set $(I,\leq)$ and a net {$F_i:i\in I$} of finite subsets of $G$ such that for every $γ ∈ G$,
...

**4**

votes

**1**answer

249 views

### left- and right- Folner sets

Given an amenable group, it is a standard trick to turn a left-invariant mean ( i.e. a continuous positive normalised linear functional $m:\ell_\infty(G) \to \mathbb{R}$ such that $\forall g \in G, m ...

**5**

votes

**0**answers

297 views

### The kernel of all invariant means

Let $G$ be discrete group which is amenable (i.e. it admits an left-invariant mean, i.e. a continuous positive normalised linear functional on $m:\ell^\infty(G) \to \mathbb{R}$ such that $\forall g ...

**5**

votes

**0**answers

205 views

### Can invariant means be really considered as the generalization of the uniform measure?

I am writing a paper for game theorists where I use (countable) amenable groups to do some things. So I am writing up a preliminary section about countable amenable groups whose main purpose is to ...

**10**

votes

**1**answer

306 views

### Subgroups of amenable periodic groups

Does every countable, infinite, amenable, periodic group $G$ contain an infinite locally finite subgroup?
Remarks:
I would be happy with an infinitely generated counterexample as long as it is ...

**14**

votes

**0**answers

442 views

### The multiplication game on the free group

Fix $W\subseteq\mathbb F_2$ and consider the following two-person game: Player 1 and Player 2 simultaneously choose $x$ and $y$ in $\mathbb F_2$. The first player wins, say one dollar, iff $xy\in W$. ...

**6**

votes

**1**answer

425 views

### Cake-cutting and amenable groups

I recently heard Alan Taylor speak about envy-free fair division and started wondering if questions like these make sense if we replace finitely additive measures with invariant means on amenable ...

**5**

votes

**0**answers

197 views

### Can a non-amenable group have a 'centrally invariant mean'?

Let $G$ be a countable, discrete group, and $f\in\ell^\infty(G)$. Let me say that $f$ has a centrally invariant mean if there is a finitely additive probability measure $\mu$ on $G\times G$ such that ...

**1**

vote

**0**answers

426 views

### Why groups that admit Folner Sequences are amenable

I've been looking at Folner's Condition recently, and I'm struggling to find a proof for why the existence of a Folner sequence on a locally compact group implies that it is amenable (and the converse ...

**1**

vote

**1**answer

190 views

### Left mean values vs right mean values

Let $G$ be a countable amenable group and $f\in\ell^\infty(G)$. Denote by $L,R,I$ respetively the sets of left-, right- and bi-invariant means on $G$. Denote by $M_L(f)$ (resp. $M_R(f),M_I(f)$) be the ...

**6**

votes

**1**answer

399 views

### A variational principle for amenable groups

Update: If somebody is interested, in Sec. 3, Theorem 3.5, of http://arxiv.org/abs/1203.2301 the variational principle for amenable groups such that every conjugacy class is finite is proved.
Let $G$ ...

**3**

votes

**1**answer

329 views

### Fubini's theorem and unique mean value

Following the terminology of Rosenblatt, I will say that a bounded function $f:\mathbb Z\rightarrow\mathbb R$ has a unique mean value if for every pair of finitely additive translation invariant ...

**6**

votes

**0**answers

395 views

### References for “folklore” on strong amenability of (group) C*-algebras?

[Apologies in advance for the prolixity - but I was unsure how much of the story is familiar.]
$\newcommand{\ptp}{\widehat{\otimes}}
\newcommand{\co}{\operatorname{co}}
...

**2**

votes

**1**answer

423 views

### Amenability of Thompson's group looking at a 4-manifold having it as the fundamental group

Just for curiosity I have done a quick web-search and I have seen that some people are studying manifolds with amenable fundamental group. On the other hand, any finitely presented group and then, in ...

**1**

vote

**1**answer

233 views

### Amenability with respect to a function

Let $(G,\cdot)$ be a group and $\phi:G\rightarrow\mathbb R$ bounded. Let me say that the pair $(G,\phi)$ is amenable if there is a finitely additive probability measure $\mu$ on $G$ such that for all ...