# Questions tagged [amenability]

Use this tag for questions related to amenable groups, which are locally compact topological groups carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements.

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### Number of maximal subgroups in finitely generated amenable groups

The following statement is known to be true: Any subgroup of a finitely generated group lies in a maximal subgroup Proof: Suppose, $G = \langle \{x_1, … , x_n\} \rangle$ is a counterexample. Then ...
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### Could we define an average that satisfied all the conditions below? If so, how?

I posted this question on Math Overflow if you wish to answer there. Consider $P:A\to\mathbb{R}$, where $A$ is a subset of $\mathbb{R}$. I want the average of $P$ to ALWAYS be between the infimum ...
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### $\{T_n\}$ Folner $\implies \{S_n\} = \{\bigcup_{k=1}^{n}T_k\}$ Folner?

Given an countable amenable group $G$, let $\{T_n\}_{n \in \mathbb{N}}$ be a Folner sequence for $G$, i.e., $\lim_{n \to +\infty} \frac{|gT_n \Delta T_n|}{|T_n|} = 0$, for every $g \in G$. Now, for ...
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### Can an integral defined by a density with respect to folner sequences of $A$ exist on a function defined on subsets of $A$?

Consider the following function where $A=\mathbb{Q}$ $$F(x)=\begin{cases} 2^x & x=A_1\\ x^2 & x=A_2\\ \text{Undefined} & \text{Everywhere Else} \end{cases}$$ A_1=\left\{\frac{2m+1}{2n+...
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### References books and lecture notes for Amenability

I am reading the book "Lectures on amenability" by Volker Runde I was wondering if someone could suggest me some books and Lecture notes with some good problems to go over My backgrounds are the ...
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### The positive net of a weak* convergent net is weak* convergent.

Suppose $X$ is a unital $C^*$-algebra, $i:X\to X^{**}$ is the natural isometric inclusion as Banach space. Denote $S_X=\{x\in X:\|x\|=1\}$, $S_{X,+}=\{x\in X:\|x\|=1,x\ge 0\}$, so does $S_{X^{**},+}$. ...
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### Polynomially sized Folner sets

Let $\Gamma$ be a finitely-generated group with a fixed finite generating set $S$. Then, $\Gamma$ is amenable if and only if it there is a sequence $(F_n)_{n=1}^{\infty}$ of finite subsets of $\Gamma$...
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### existence of weak* continuous means for amenable locally compact groups

Let G be an amenable locally compact group. Does there exist a left invariant mean $m \colon L^\infty(G) \to \mathbb{C}$ on $L^\infty(G)$ which is in addition weak*-continuous ? Recall that the von ...
I have the following statement: if all groups in the directed system $\{G_{i}\}_{i\in I}$ are amenable, then so is their directed union $G:= \bigcup_{i\in I}G_{i}$ (Remember that a group is amenable ...