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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.

An amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements.

The amenability property has a large number of equivalent formulations. In the field of analysis, the definition is in terms of linear functionals. An intuitive way to understand that version is that the support of the regular representation is the whole space of irreducible representations.

In discrete group theory, where G has the discrete topology, a simpler definition is used: A group is amenable if one can say what proportion of G any given subset takes up.

If a group has a Følner sequence, then the group is automatically amenable.

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Definition of amenable group

I have made several attempts at various times to understand the many equivalent definitions of an amenable group. Is the following statement correct? A group $G$ is amenable if and only if, for any finite subset $X$ of $G$ and any $\epsilon > 0$,…
Derek Holt
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Derivation into dense ideal of Banach algebras

Let $A$ be a Banach algebra and $I$ be an ideal of $A$. A derivation $D\colon A\to I$ is a linear bounded map, with the following property: $$D(ab)=aD(b)+D(a)b,\qquad a,b\in A.$$ Suppose that $I$ is dense in $A$, and any derivation $D\colon A\to I$…
Hamid Shafie Asl
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How strong is the statement that Thompson $F$ is amenable?

Justin Moore's proof turned out to have an error I just attended Justin Moore's talk on this today. Since I am neither a group theorist nor a combinatorist, and is not familiar with ultrafilters I cannot judge the correctness of his proof. But more…
Bombyx mori
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Nonamenable subgroups of the unitary group of the hyperfinite II_1 factor

The hyperfinite $II_1$ factor arises as the group von Neumann algebra of any infinite amenable group such that every conjugacy class but that of the identity has infinite cardinality. The unitary group of this von Neumann algebra contains every…
Jon Bannon
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Why is "Amenable Group" a pun?

"The original definition, in terms of a finitely additive invariant measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In…
Bananach
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Problems with Set Function

(Questions are at the bottom of the post. I’ve added two more doubts since the recent answer.) Introduction Consider $f:A\to\mathbb{R}$ where $A\subseteq[a,b]$, $a,b \in \mathbb{R}$ and $S$ is a fixed subset of $A$. Before mentioning my set…
Arbuja
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How do probability measures on $\mathbb{Z}$ look like?

Let $\mu: \mathcal{P}(\mathbb{Z}) \rightarrow [0,1]$ be a finitely additive $\mathbb{Z}$-invariant probability measure on $\mathbb{Z}$. Such measures exist because $\mathbb{Z}$ is amenable and one can indeed describe examples of such measures as…
Burak
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Is $SO_2$ an amenable group?

In S. Wagon's "The Banach-Tarski Paradox," amenable groups are defined on p. 12 as follows: [amenable] groups bear a left-invariant, finitely additive measure of total measure one that is defined on all subsets. He defines $SO_2$ to be the group…
Rachel
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Is there an example of a non compact, semisimple, amenable Lie group?

By semisimple I mean the real Lie algebra of $G$ is semisimple. I guess there is not but I can't formulate a rigorous argument.
Ariosto
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von Neumann algebra associated to the full group C*-algebra

Lance's theorem asserts that a discrete group $G$ is amenable if and only if the reduced and full groups C*-algebras coincide. The group von Neumann algebra is the weak closure of the reduced group C*-algebra concretely represented on $\ell_2(G)$.…
Tomasz Kania
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How big is a class of groups with a certain strong amenability condition?

Let $G$ be a countably infinite discrete group. Let us call $G$ sequentially amenable, if there is a bijection $\mathbb{N}\to G,\; j\mapsto g_j$ such that the sets $F_k = \{g_1,\dots,g_k\}$ form a Folner sequence for $G$. How big is the class of…
Gabor Szabo
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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 each $n \in \mathbb{N}$, consider $S_n =…
Luísa Borsato
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Is there a connection between my density formula and an invariant mean defined by a folner sequence of rational numbers?

I am a first-year undergraduate who stumbled upon natural density. I am working on extending this definition to the subset of rational numbers. While most people would wait until they are older, I am already attempting to solve the problem. I sent a…
Arbuja
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How is this property equivalent to the Reiter Property?

We have the Reiter Property $(R_2)$ for an action of a group G on a set X: For any $\epsilon>0$, any finite subset $S$ of G, there exists $\phi\in{\ell^2(X)}$ such that $\|s\phi-\phi\|_{\ell^2}<\epsilon{\|\phi\|_{\ell^2}}$ for all $s\in{S}$. I am…
user294388
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Is the von neumann algebra of locally compact amenable group hyperfinite?

Let $G$ be a discrete group and $\mathcal{L}(G)$ the associated von Neumann algebra. It is well known that $G$ is amenable if and only if $\mathcal{L}(G)$ is hyperfinite. Does there exist a generalization of this theorem to arbitrary locally compact…
Zouba
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