39.05.06 · operator-algebras / nuclearity-exactness

Amenable Groups: Invariant Means, the Følner Condition, and Paradoxical Decompositions

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Anchor (Master): von Neumann 1929 Fund. Math. 13; Følner 1955 Math. Scand. 3; Day 1957 Illinois J. Math. 1; Ol'shanskii 1980; Brown-Ozawa Ch. 2; Wagon, The Banach-Tarski Paradox

Intuition Beginner

Suppose you want to take a fair average over the whole numbers — positive, negative, and zero. For a finite list you just add the values and divide by how many there are. But the integers go on forever in both directions, so there is no count to divide by. Worse, you would like the average to ignore where you start: shifting every number up by one should not change the average of a function on the integers. Can such a shift-proof average exist at all?

For the integers the answer is yes, in a generous sense. You can average over a huge block of numbers, then over an even huger block, and the answers settle down to something stable that does not care about shifting. A group for which this kind of fair, shift-proof average exists is called amenable. The word was coined to suggest a group that is "able to be a-mean-ed" — a small mathematician's pun on mean, the word for an average.

Not every group is so cooperative. There are groups so richly branched that you can chop them into a few pieces, slide the pieces around using the group's own moves, and reassemble them into two complete copies of the original group. A group that allows such a duplication is called paradoxical, and no fair average can survive it: averaging the one original would have to equal averaging the two copies, which is impossible. The simplest paradoxical group is the free group on two letters, the group of all words you can spell with $a$ , $b$ , and their inverses with no relations. This same duplication trick, pushed into three-dimensional space, is what powers the famous Banach-Tarski "pea into the sun" result.

Why does this matter beyond a curiosity? Amenability is exactly the dividing line that controls whether the algebra of operators built from a group is well-behaved enough to be approximated by finite-dimensional pieces.

Visual Beginner

The picture contrasts two groups. On the left, the integers are drawn as a single line of evenly spaced dots; a sliding window (a long block of dots) can be made as long as you like, and shifting the whole line by one barely changes which dots the window covers. This "barely changes under shifting" is the averaging property. On the right, the free group on two letters is drawn as an infinitely branching tree, four branches at every dot; any window you draw has almost all of its dots on the boundary, because the tree grows so fast that the rim always rivals the interior. That fast growth is what blocks a fair average and permits the duplication trick.

The contrast is between linear growth (a window of length $n$ has $2$ boundary points) and exponential growth (a ball of radius $n$ in the tree has a boundary as large as the whole ball).

Worked example Beginner

Take the group of integers and the finite blocks $F_{n} = {- n, - n + 1, \dots, n - 1, n}$ , each holding $2 n + 1$ numbers. We check that shifting $F_{n}$ by one barely disturbs it, and that the disturbance shrinks to nothing as $n$ grows.

Step 1. Shift $F_{n}$ up by $1$ to get $F_{n} + 1 = {- n + 1, \dots, n + 1}$ . Compare it with $F_{n}$ . The two blocks agree on ${- n + 1, \dots, n}$ and differ only at the two endpoints: the number $- n$ is in $F_{n}$ but not in the shifted block, and $n + 1$ is in the shifted block but not in $F_{n}$ .

Step 2. Count the mismatch. Exactly $2$ numbers fail to match between $F_{n}$ and its shift.

Step 3. Compare the mismatch to the size of the block. The block has $2 n + 1$ numbers, so the fraction that fails to match is $\frac{2}{2 n + 1}$ . For $n = 10$ this is $\frac{2}{21}$ , about $0.095$ . For $n = 1000$ it is $\frac{2}{2001}$ , about $0.001$ .

Step 4. Let the blocks grow. As $n$ gets large, $\frac{2}{2 n + 1}$ heads to $0$ . The bigger the block, the more nearly a shift leaves it unchanged.

What this tells us: there is a sequence of finite blocks that become more and more shift-proof. Averaging a function over these growing blocks gives a fair, shift-resistant average. That is precisely what makes the integers an amenable group.

Check your understanding Beginner

Exercise (easy, multiple choice).

A group is called amenable when it admits which of the following?

A. A way to cut it into pieces and reassemble two copies of itself

B. A fair, shift-resistant way to average functions on the group

C. A finite number of elements

D. A relation among its generators

Hint

The name puns on the word for an average. Which option describes an averaging operation that respects the group's own shifts?

Answer

B. Amenability means there is an invariant (shift-resistant) average, a "mean," on functions on the group. Feedback-correct: this is exactly the averaging property the integers enjoy through growing blocks. Feedback-wrong: option A describes a paradoxical group, which is the opposite of amenable; finiteness (C) is sufficient but not the definition; and a relation among generators (D) is unrelated.

Formal definition Intermediate+

Throughout, $G$ is a discrete group and $ℓ^{\infty} (G)$ is the Banach space of bounded complex-valued functions on $G$ with the supremum norm. The group acts on $ℓ^{\infty} (G)$ by left translation: $(s \cdot f) (t) = f (s^{- 1} t)$ for $s, t \in G$ .

Definition (invariant mean). A mean on $G$ is a linear functional $m : ℓ^{\infty} (G) \to C$ that is positive ( $m (f) \geq 0$ whenever $f \geq 0$ ) and unital ( $m (1) = 1$ ); equivalently $m$ is a state on $ℓ^{\infty} (G)$ with $∥ m ∥ = m (1) = 1$ . The mean is left-invariant if $m (s \cdot f) = m (f)$ for all $s \in G$ and $f \in ℓ^{\infty} (G)$ . The group $G$ is amenable if it admits a left-invariant mean ^{[Brown-Ozawa Ch. 2]}. A left-invariant mean is a finitely additive, translation-invariant probability measure defined on all subsets of $G$ via $μ (A) = m (1_{A})$ .

Definition (Følner condition). $G$ satisfies the Følner condition if there is a net $(F_{i})_{i \in I}$ of nonempty finite subsets of $G$ such that for every $s \in G$ , $i lim \frac{∣ s F _{i} △ F _{i} ∣}{∣ F _{i} ∣} = 0,$ where $△$ is symmetric difference. When $G$ is countable a sequence $(F_{n})_{n \geq 1}$ suffices, and one calls it a Følner sequence. An equivalent boundary formulation: writing $\partial_{s} F = (s F △ F)$ , the sets $F_{i}$ have asymptotically negligible translation-boundary.

Definition (Reiter's condition). $G$ satisfies Reiter's property $(P_{1})$ if there is a net of unit vectors $ξ_{i}$ in the probability simplex $Prob (G) \subseteq ℓ^{1} (G)$ (so $ξ_{i} \geq 0$ , $\sum_{t} ξ_{i} (t) = 1$ ) with $∥ s \cdot ξ_{i} - ξ_{i} ∥_{1} \to 0$ for every $s \in G$ . The normalised indicators $ξ_{i} = ∣ F_{i} ∣^{- 1} 1_{F_{i}}$ of a Følner net realise Reiter's property.

Definition (fixed-point property). $G$ has the fixed-point property if every continuous affine action of $G$ on a nonempty compact convex subset $K$ of a locally convex topological vector space has a fixed point ^{[Day 1957]}.

Definition (paradoxical decomposition). $G$ is paradoxical if there exist a partition $G = ⨆_{i = 1}^{n} A_{i} ⊔ ⨆_{j = 1}^{m} B_{j}$ into finitely many disjoint pieces and group elements $g_{1}, \dots, g_{n}, h_{1}, \dots, h_{m} \in G$ such that $G = i = 1 ⨆ n g_{i} A_{i} = j = 1 ⨆ m h_{j} B_{j} .$ That is, the $A_{i}$ alone, translated, retile $G$ , and the $B_{j}$ alone, translated, also retile $G$ : the group is cut into finitely many pieces that reassemble into two copies of itself.

Counterexamples to common slips Intermediate+

Invariant mean is not a countably additive measure. The functional $m$ is only finitely additive. There is no countably additive translation-invariant probability measure on all of $Z$ — the singletons would each get measure $0$ , summing to $0 \neq = 1$ . Finite additivity is what makes amenability attainable.
Følner sets need not be balls or nested. The condition constrains only the asymptotic translation-boundary ratio. For $Z^{2}$ long thin rectangles fail (their boundary ratio does not vanish) while squares succeed; the existence of some Følner net is what is required, not that every natural exhaustion works.
Non-abelian does not mean non-amenable. The infinite dihedral group and every solvable group are amenable despite being non-abelian. Non-amenability requires the kind of exponential branching present in a free subgroup, not mere non-commutativity.
A paradoxical decomposition uses finitely many pieces. Allowing countably many pieces empties the notion of content (one can always split $Z$ countably and reshuffle); the force of Tarski's theorem is precisely the finite piece count.

Key theorem with proof Intermediate+

Theorem (Følner $\Rightarrow$ invariant mean). If $G$ satisfies the Følner condition, then $G$ admits a left-invariant mean; hence $G$ is amenable ^{[Følner 1955]}.

Proof. Let $(F_{i})_{i \in I}$ be a Følner net. For each $i$ define $m_{i} \in ℓ^{\infty} (G)^{*}$ by averaging over $F_{i}$ : $m_{i} (f) = \frac{1}{∣ F _{i} ∣} t \in F_{i} \sum f (t) .$ Each $m_{i}$ is positive and unital, so it is a state on $ℓ^{\infty} (G)$ . The state space $S$ of $ℓ^{\infty} (G)$ is weak*-compact by the Banach-Alaoglu theorem, since it is a weak*-closed subset of the unit ball of $ℓ^{\infty} (G)^{*}$ . Therefore the net $(m_{i})$ has a weak*-convergent subnet $(m_{i_{k}})$ with limit $m \in S$ . The limit $m$ is again positive and unital, hence a mean.

It remains to show $m$ is left-invariant. Fix $s \in G$ and $f \in ℓ^{\infty} (G)$ with $∥ f ∥_{\infty} \leq 1$ . Compute the displacement of $m_{i}$ under the translation $s$ : $$ m_i(s \cdot f) - m_i(f) = \frac{1}{|F_i|} \sum_{t \in F_i} \big( f(s^{-1} t) - f(t) \big) = \frac{1}{|F_i|} \left( \sum_{u \in s^{-1} F_i} f(u) - \sum_{t \in F_i} f(t) \right). $$ The two sums agree on $s^{- 1} F_{i} \cap F_{i}$ and differ only on the symmetric difference $s^{- 1} F_{i} △ F_{i}$ . Each term is bounded by $∥ f ∥_{\infty} \leq 1$ in absolute value, so $$ \big| m_i(s \cdot f) - m_i(f) \big| \leq \frac{| s^{-1} F_i \triangle F_i |}{|F_i|}. $$ Because $∣ s^{- 1} F_{i} △ F_{i} ∣ = ∣ F_{i} △ s F_{i} ∣$ (apply the bijection $t \mapsto s t$ ), the Følner condition forces the right-hand side to $0$ along the net. Passing to the convergent subnet, $m (s \cdot f) - m (f) = 0$ . Since $s$ and $f$ were arbitrary, $m$ is left-invariant. $□$

Bridge. This theorem is the foundational reason the combinatorial Følner picture and the analytic invariant-mean picture describe the same class of groups: the normalised indicator of a Følner set is an approximately invariant probability vector, and weak*-compactness of the state space converts approximate invariance into exact invariance in the limit. This is exactly the mechanism by which a constructive averaging procedure produces a non-constructive functional. The converse — that an invariant mean yields Følner sets — is harder and runs through Reiter's property; putting these together gives the von Neumann-Day equivalence. The bridge is between geometry (asymptotically negligible boundary) and functional analysis (a translation-invariant state), and it builds toward the master-tier closure theorems, where each closure property of the amenable class becomes a manipulation of Følner sets, and it appears again in the proof that $C_{r}^{*} (G)$ is nuclear precisely when $G$ is amenable, where the Følner sets supply the approximating finite-dimensional structure.

Exercises Intermediate+

Exercise 4 (medium, symbolic).

Show that $Z^{2}$ is amenable by exhibiting an explicit Følner sequence and bounding its Følner ratios for the standard generators.

Hint

Use square boxes $F_{n} = {- n, \dots, n}^{2}$ and count how many points a unit shift moves out of the box.

Answer

Let $F_{n} = {- n, \dots, n}^{2}$ , so $∣ F_{n} ∣ = (2 n + 1)^{2}$ . Take $s = (1, 0)$ . Then $s F_{n} △ F_{n}$ consists of the two vertical edge-columns of width one, namely ${- n - 1} \times {- n, \dots, n}$ added and ${n} \times {- n, \dots, n}$ shifted away after intersecting; precisely $∣ s F_{n} △ F_{n} ∣ = 2 (2 n + 1)$ . Hence $\frac{∣ s F _{n} △ F _{n} ∣}{∣ F _{n} ∣} = \frac{2 ( 2 n + 1 )}{( 2 n + 1 ) ^{2}} = \frac{2}{2 n + 1} \to 0.$ By symmetry the same bound holds for $s = (0, 1)$ , and any group element is a product of generators, so its boundary ratio is controlled by the triangle inequality $∣ s s^{'} F △ F ∣ \leq ∣ s s^{'} F △ s^{'} F ∣ + ∣ s^{'} F △ F ∣ = ∣ s F △ F ∣ + ∣ s^{'} F △ F ∣$ . Thus $(F_{n})$ is a Følner sequence and $Z^{2}$ is amenable.

Exercise 5 (medium, symbolic).

Prove that if $G$ is amenable and $H \leq G$ is a subgroup of finite index, then a Følner sequence for $G$ can be used to produce a left-invariant mean restricted to $H$ -translation. (Establish the easy direction: a finite-index subgroup of an amenable group is amenable, via the mean.)

Hint

Restrict an invariant mean on $G$ to functions supported by coset structure, or push functions on $H$ up to $G$ by extending by a coset average. Use that $H$ -translation is a special case of $G$ -translation.

Answer

Let $m$ be a left-invariant mean on $ℓ^{\infty} (G)$ . Choose right-coset representatives $g_{1}, \dots, g_{k}$ for $H$ in $G$ (finite since $[G : H] = k$ ). Given $f \in ℓ^{\infty} (H)$ , define $\tilde{f} \in ℓ^{\infty} (G)$ by $\tilde{f} (g) = f (h)$ where $g = h g_{j}$ is the unique factorisation with $h \in H$ ; that is, push $f$ along the projection $G \to H$ . Set $m_{H} (f) = m (\tilde{f})$ . Linearity, positivity, and unitality of $m_{H}$ are inherited from $m$ since $1 = 1$ and pushing preserves order. For $h_{0} \in H$ , the map $g \mapsto h_{0}^{- 1} g$ sends the $H$ -component $h$ to $h_{0}^{- 1} h$ while fixing the coset representative, so $h_{0} \cdot f = h_{0} \cdot \tilde{f}$ ; invariance of $m$ gives $m_{H} (h_{0} \cdot f) = m (h_{0} \cdot \tilde{f}) = m (\tilde{f}) = m_{H} (f)$ . Hence $m_{H}$ is a left-invariant mean on $H$ and $H$ is amenable.

Exercise 6 (medium, multiple choice).

Reiter's property $(P_{1})$ asks for a net of probability vectors $ξ_{i} \in Prob (G)$ with which property?

A. $∥ ξ_{i} ∥_{2} \to 0$

B. $∥ s \cdot ξ_{i} - ξ_{i} ∥_{1} \to 0$ for every $s \in G$

C. $ξ_{i} \to 0$ pointwise

D. $⟨ s \cdot ξ_{i}, ξ_{i} ⟩ \to 0$ for every $s \in G$

Hint

Reiter's vectors are approximately invariant probability densities; the relevant norm is the one in which probability measures live.

Answer

B. Reiter's property requires approximately invariant unit vectors in $ℓ^{1}$ : $∥ s \cdot ξ_{i} - ξ_{i} ∥_{1} \to 0$ for all $s$ . Feedback-wrong: option A and C describe vectors decaying to zero (incompatible with being probability vectors of total mass one); option D is the kind of orthogonality/asymptotic-invariance condition associated with the Kazhdan property (T) or with weak containment in the regular representation, not Reiter's $(P_{1})$ .

Exercise 7 (hard, symbolic).

Prove that the free group $F_{2} = ⟨ a, b ⟩$ is paradoxical, and deduce that it is non-amenable.

Hint

Partition $F_{2} ∖ {e}$ by the first letter of the reduced word into $W (a), W (a^{- 1}), W (b), W (b^{- 1})$ . Show $a^{- 1} W (a^{- 1})$ fills out a large part of $F_{2}$ , and similarly for $b$ . Recall (from the free-group unit) that reduced words are unique.

Answer

By uniqueness of reduced words 01.02.20, every nonidentity element of $F_{2}$ begins with exactly one of $a, a^{- 1}, b, b^{- 1}$ . Let $W (x)$ be the set of reduced words starting with $x$ . Then $F_{2} = {e} ⊔ W (a) ⊔ W (a^{- 1}) ⊔ W (b) ⊔ W (b^{- 1}) .$ Claim: $F_{2} = W (a) ⊔ aW (a^{- 1})$ . Indeed, if a reduced word $w$ does not start with $a$ , then $a^{- 1} w$ is reduced and starts with $a^{- 1}$ (no cancellation, since $w$ 's first letter is not $a$ ), so $w = a (a^{- 1} w) \in aW (a^{- 1})$ ; and if $w$ starts with $a$ it is in $W (a)$ . The two sets are disjoint because elements of $W (a)$ start with $a$ while elements of $aW (a^{- 1})$ , being $a \cdot (word starting a^{- 1})$ , start with whatever survives the cancellation and never start with $a$ . Symmetrically $F_{2} = W (b) ⊔ bW (b^{- 1})$ .

Now $G = F_{2}$ is covered twice: the pieces $W (a)$ and $W (a^{- 1})$ (translated by $e$ and $a$ ) reassemble all of $F_{2}$ , and the pieces $W (b)$ and $W (b^{- 1})$ (translated by $e$ and $b$ ) also reassemble all of $F_{2}$ . Absorbing ${e}$ into one piece, this is a paradoxical decomposition. If $F_{2}$ had a left-invariant finitely additive probability measure $μ$ , then $1 = μ (F_{2}) = μ (W (a)) + μ (aW (a^{- 1})) = μ (W (a)) + μ (W (a^{- 1}))$ , and likewise $1 = μ (W (b)) + μ (W (b^{- 1}))$ . Adding gives $2 = μ (W (a)) + μ (W (a^{- 1})) + μ (W (b)) + μ (W (b^{- 1})) \leq μ (F_{2}) = 1$ , a contradiction. Hence no invariant mean exists and $F_{2}$ is non-amenable.

Exercise 8 (hard, symbolic).

Prove that an extension of amenable groups is amenable: if $1 \to N \to G \to Q \to 1$ is exact with $N$ and $Q$ amenable, then $G$ is amenable. (Use invariant means; you may assume means on $N$ and $Q$ exist.)

Hint

Given $f \in ℓ^{\infty} (G)$ , average over $N$ first to get a function constant on $N$ -cosets, hence a function on $Q$ ; then average that over $Q$ . Check each averaging step is invariant.

Answer

Let $m_{N}$ be a left-invariant mean on $ℓ^{\infty} (N)$ and $m_{Q}$ on $ℓ^{\infty} (Q)$ ; write $π : G \to Q$ for the quotient map. Given $f \in ℓ^{\infty} (G)$ and $g \in G$ , the function $n \mapsto f (g n)$ lies in $ℓ^{\infty} (N)$ ; define $(P f) (g) = m_{N} (n \mapsto f (g n))$ . Left $N$ -invariance of $m_{N}$ makes $P f$ constant on left cosets $g N$ : for $n_{0} \in N$ , $(P f) (g n_{0}) = m_{N} (n \mapsto f (g n_{0} n)) = m_{N} (n_{0} \cdot (n \mapsto f (g n))) = (P f) (g)$ . Hence $P f$ descends to a bounded function $\overline{P f} \in ℓ^{\infty} (Q)$ with $\overline{P f} (π (g)) = (P f) (g)$ . Define $m (f) = m_{Q} (\overline{P f})$ .

The functional $m$ is positive and unital because $P$ and $m_{N}, m_{Q}$ are. For left-invariance, fix $g_{0} \in G$ . A direct computation using right-translation bookkeeping inside $m_{N}$ shows $\overline{P (g_{0} \cdot f)} = π (g_{0}) \cdot \overline{P f}$ : averaging $f (g_{0}^{- 1} g n)$ over $n \in N$ produces the value of $P f$ at $g_{0}^{- 1} g$ , which as a function on $Q$ is the $π (g_{0})$ -translate of $\overline{P f}$ . Then left-invariance of $m_{Q}$ gives $m (g_{0} \cdot f) = m_{Q} (π (g_{0}) \cdot \overline{P f}) = m_{Q} (\overline{P f}) = m (f)$ . Thus $m$ is a left-invariant mean and $G$ is amenable.

Advanced results Master

Theorem 1 (von Neumann-Day equivalences). For a discrete group $G$ the following are equivalent: (i) $G$ has a left-invariant mean on $ℓ^{\infty} (G)$ ; (ii) $G$ satisfies the Følner condition; (iii) $G$ satisfies Reiter's property $(P_{1})$ ; (iv) $G$ has the fixed-point property for affine actions on compact convex sets; (v) the one-dimensional unit representation is weakly contained in the left regular representation $λ$ (Hulanicki's criterion); (vi) the reduced group C*-algebra $C_{r}^{*} (G)$ coincides with the full group C*-algebra $C^{*} (G)$ ^{[Brown-Ozawa Ch. 2]}. The cycle (ii) $\Rightarrow$ (i) is the Key theorem; (i) $\Rightarrow$ (iv) is the Day fixed-point argument; (iv) $\Rightarrow$ (i) applies the fixed-point property to the affine action on the weak*-compact convex set of means; (iii) $\Leftrightarrow$ (ii) passes between Følner indicators and $ℓ^{1}$ densities.

Theorem 2 (closure properties). The class of amenable groups is closed under: (a) passage to subgroups; (b) quotients; (c) extensions; (d) directed unions (direct limits). Consequently every solvable group is amenable (built from abelian groups by iterated extension), every locally finite group is amenable, and more generally the smallest class containing all finite and abelian groups and closed under these four operations — the class of elementary amenable groups (Day) — consists of amenable groups ^{[Day 1957]}.

Theorem 3 (growth and amenability). Every group of subexponential growth is amenable: if $γ_{G} (n) = ∣ B (e, n) ∣$ (the ball of radius $n$ in a word metric) satisfies $lim_{n} γ_{G} (n)^{1/ n} = 1$ , then the balls $B (e, n)$ , or a subsequence of them, form a Følner sequence (Adelson-Velskii-Šreider). In particular all groups of polynomial growth — by Gromov's theorem, exactly the virtually nilpotent groups — are amenable. The converse fails: there exist amenable groups of exponential growth, such as the lamplighter group $Z /2 ≀ Z$ and solvable Baumslag-Solitar groups $B S (1, n)$ .

Theorem 4 (Tarski's theorem). A group $G$ is non-amenable if and only if it admits a paradoxical decomposition. Equivalently, $G$ is amenable if and only if it is not paradoxical ^{[Tarski 1938]}. More sharply, Tarski's invariant: there is a finitely additive left-invariant probability measure on $G$ if and only if $G$ is not $G$ -paradoxical, and the obstruction is detected by the Tarski number, the least number of pieces in a paradoxical decomposition (which is $\geq 4$ , with $4$ achieved exactly by groups containing $F_{2}$ in the relevant sense).

Theorem 5 (Banach-Tarski connection). The free group $F_{2}$ embeds in $S O (3)$ as a subgroup acting freely on the sphere $S^{2}$ minus a countable set; its paradoxical decomposition transports through this free action to a paradoxical decomposition of $S^{2}$ , and thence of the solid ball in $R^{3}$ , using the axiom of choice to select orbit representatives. The result — a solid ball cut into finitely many pieces and reassembled by rigid motions into two balls of the same radius — is impossible in dimensions $1$ and $2$ precisely because $S O (2)$ and the isometry groups of $R^{1}, R^{2}$ are amenable, so a finitely additive isometry-invariant measure exists and forbids duplication ^{[Wagon 1985]}.

Theorem 6 (von Neumann conjecture and Ol'shanskii's counterexample). Von Neumann's 1929 work isolated the free group $F_{2}$ as the source of paradoxicality, and the von Neumann conjecture (attributed to him, formalised by Day) asserted that a group is non-amenable if and only if it contains a copy of $F_{2}$ . The conjecture is false: Ol'shanskii constructed in 1980 a non-amenable group all of whose proper subgroups are cyclic (a Tarski monster), which therefore contains no copy of $F_{2}$ ^{[Ol'shanskii 1980]}. Adian had earlier shown the free Burnside groups $B (m, n)$ are non-amenable for large odd $n$ , $m \geq 2$ , and these are torsion groups containing no $F_{2}$ . A finitely presented counterexample was later given by Ol'shanskii-Sapir.

Theorem 7 (Kesten's spectral criterion). Let $μ$ be a symmetric probability measure on $G$ whose support generates $G$ , and let $λ (μ) = \sum_{g} μ (g) λ (g)$ be the associated Markov (convolution) operator on $ℓ^{2} (G)$ . Then $G$ is amenable if and only if the operator norm $∥ λ (μ) ∥ = 1$ (equivalently, the spectral radius of the random walk equals $1$ ); for non-amenable $G$ the norm is strictly less than $1$ , giving exponential decay of return probabilities. This is the analytic shadow of the Følner condition: Følner sets are approximate eigenvectors of $λ (μ)$ with eigenvalue near $1$ ^{[Brown-Ozawa Ch. 2]}.

Theorem 8 (amenability and nuclearity). For a discrete group $G$ , the reduced group C*-algebra $C_{r}^{*} (G)$ is nuclear if and only if $G$ is amenable; moreover amenability of $G$ implies $C_{r}^{*} (G)$ is exact, and the group von Neumann algebra $L (G)$ is hyperfinite (injective) if and only if $G$ is amenable (Connes' theorem in the ICC case). The Følner sequence supplies the completely positive finite-rank approximations $φ_{n} : C_{r}^{*} (G) \to M_{∣ F_{n} ∣} (C) \to C_{r}^{*} (G)$ witnessing the completely positive approximation property ^{[Brown-Ozawa Ch. 2]}.

Synthesis. Amenability is the central insight that unifies four superficially unrelated theories: the analytic theory of invariant means, the combinatorial-geometric theory of Følner sets, the measure-theoretic theory of paradoxical decompositions, and the operator-algebraic theory of nuclearity. The von Neumann-Day equivalence is exactly the statement that these descriptions coincide, and the foundational reason they coincide is weak*-compactness of the state space, which converts the approximate invariance of Følner indicators into an exact invariant functional. Putting these together with Tarski's theorem, amenability is dual to paradoxicality: a group either carries a finitely additive invariant probability measure or it duplicates itself, never both, and this dichotomy is what the Banach-Tarski paradox exploits once a free subgroup of $S O (3)$ is available. The bridge from group theory to operator algebras is Theorem 8: Følner sets are the approximating finite-dimensional structure that makes $C_{r}^{*} (G)$ nuclear, so the group-theoretic dividing line becomes the operator-algebraic one, and this generalises from groups to group actions and groupoids, where amenability of the action is exactly the input needed for nuclearity of the crossed product. The failure of the von Neumann conjecture — Ol'shanskii's Tarski monsters and Adian's Burnside groups — shows the dichotomy is genuinely subtler than "contains $F_{2}$ or not," yet the equivalence of means, Følner sets, and nuclearity survives intact.

Full proof set Master

Proposition 1 (subgroups of amenable groups are amenable). Let $G$ be amenable and $H \leq G$ . Then $H$ is amenable.

Proof. Let $m$ be a left-invariant mean on $ℓ^{\infty} (G)$ . Choose, by the axiom of choice, a transversal $T \subseteq G$ for the right cosets $H \ G$ , so that $G = ⨆_{t \in T} H t$ and every $g \in G$ factors uniquely as $g = h (g) t (g)$ with $h (g) \in H$ , $t (g) \in T$ . Given $f \in ℓ^{\infty} (H)$ , define its lift $\tilde{f} \in ℓ^{\infty} (G)$ by $\tilde{f} (g) = f (h (g))$ . Set $m_{H} (f) = m (\tilde{f})$ .

Positivity and unitality of $m_{H}$ follow from those of $m$ together with $\tilde{1} = 1$ and the order-preservation $f \geq 0 \Rightarrow \tilde{f} \geq 0$ . For invariance, fix $h_{0} \in H$ . Left multiplication by $h_{0}$ sends $g = h (g) t (g)$ to $h_{0} g = (h_{0} h (g)) t (g)$ , so the transversal component is unchanged and the $H$ -component is left-translated: $h_{0} \cdot f (g) = (h_{0} \cdot f) (h (g)) = f (h_{0}^{- 1} h (g)) = \tilde{f} (h_{0}^{- 1} g) = (h_{0} \cdot \tilde{f}) (g)$ . Hence $m_{H} (h_{0} \cdot f) = m (h_{0} \cdot \tilde{f}) = m (\tilde{f}) = m_{H} (f)$ by invariance of $m$ . Thus $m_{H}$ is a left-invariant mean and $H$ is amenable. $□$

Proposition 2 (quotients of amenable groups are amenable). Let $G$ be amenable and $N ⊴ G$ with quotient $Q = G / N$ and projection $π : G \to Q$ . Then $Q$ is amenable.

Proof. Let $m$ be a left-invariant mean on $ℓ^{\infty} (G)$ . The map $π^{*} : ℓ^{\infty} (Q) \to ℓ^{\infty} (G)$ , $(π^{*} φ) (g) = φ (π (g))$ , is a unital positive linear isometry intertwining translation: $π^{*} (π (g_{0}) \cdot φ) = g_{0} \cdot π^{*} φ$ , since $(π^{*} (π (g_{0}) \cdot φ)) (g) = φ (π (g_{0})^{- 1} π (g)) = φ (π (g_{0}^{- 1} g)) = (π^{*} φ) (g_{0}^{- 1} g)$ . Define $m_{Q} (φ) = m (π^{*} φ)$ . Then $m_{Q}$ is positive and unital, and for $g_{0} \in G$ , $m_{Q} (π (g_{0}) \cdot φ) = m (π^{*} (π (g_{0}) \cdot φ)) = m (g_{0} \cdot π^{*} φ) = m (π^{*} φ) = m_{Q} (φ) .$ Since $π$ is surjective every element of $Q$ has the form $π (g_{0})$ , so $m_{Q}$ is left-invariant. Hence $Q$ is amenable. $□$

Proposition 3 (directed unions of amenable groups are amenable). Let $G = ⋃_{i \in I} G_{i}$ be a directed union of amenable subgroups $G_{i}$ (so for all $i, j$ there is $k$ with $G_{i}, G_{j} \subseteq G_{k}$ ). Then $G$ is amenable.

Proof. For each $i$ let $m_{i}$ be a left-invariant mean on $ℓ^{\infty} (G_{i})$ . Fix a transversal $T_{i}$ for the right cosets $G_{i} \ G$ and lift $m_{i}$ to a state $\overset{m}{^}_{i}$ on $ℓ^{\infty} (G)$ by $\overset{m}{^}_{i} (f) = m_{i} (g \mapsto f ∣_{G_{i}})$ after collapsing along $T_{i}$ exactly as in Proposition 1; concretely $\overset{m}{^}_{i} (f) = m_{i} (x \mapsto f (xτ))$ averaged over the transversal class, giving a state that is invariant under left translation by elements of $G_{i}$ . The states $\overset{m}{^}_{i}$ lie in the weak*-compact state space $S$ of $ℓ^{\infty} (G)$ . Order $I$ by the directed relation; the net $(\overset{m}{^}_{i})_{i \in I}$ has a weak*-cluster point $m \in S$ . For any $s \in G$ there is $i_{0}$ with $s \in G_{i_{0}}$ ; for all $i \geq i_{0}$ (in the directed order, so $G_{i} \supseteq G_{i_{0}} ∋ s$ ) the state $\overset{m}{^}_{i}$ is invariant under left translation by $s$ . Passing to the cluster point, $m (s \cdot f) = m (f)$ . As $s$ was arbitrary, $m$ is a left-invariant mean and $G$ is amenable. $□$

Proposition 4 (abelian groups are amenable — Markov-Kakutani route). Every abelian group $G$ is amenable.

Proof. The set $M$ of means on $ℓ^{\infty} (G)$ is a nonempty weak*-compact convex subset of $ℓ^{\infty} (G)^{*}$ . For each $s \in G$ , the adjoint of left translation, $L_{s}^{*} : M \to M$ given by $(L_{s}^{*} m) (f) = m (s \cdot f)$ , is an affine weak*-continuous self-map of $M$ . Because $G$ is abelian, the family ${L_{s}^{*} : s \in G}$ is commuting. By the Markov-Kakutani fixed-point theorem, a commuting family of continuous affine self-maps of a compact convex set has a common fixed point $m \in M$ . A common fixed point satisfies $m (s \cdot f) = m (f)$ for all $s$ and $f$ , i.e. $m$ is a left-invariant mean. Hence $G$ is amenable. (Combined with Proposition 2 and the extension result, this gives amenability of all solvable groups.) $□$

Connections Master

Free group, free product, group presentation 01.02.20. The free group $F_{2}$ is the canonical non-amenable group, and its paradoxical decomposition is read directly off the uniqueness of reduced words established there. The foundational reason free groups break amenability is the exponential branching of their Cayley tree, which forces every finite set to be almost all boundary, so no Følner sequence exists; this is dual to the integers, whose linear Cayley graph makes intervals Følner. Free products of nontrivial groups (except $Z /2 * Z /2$ ) inherit a copy of $F_{2}$ and are non-amenable, tying the free-product construction directly to the amenability dichotomy.
Sigma-algebra, measurable space, and the Borel sigma-algebra 02.07.01. An invariant mean is a finitely additive translation-invariant probability defined on the entire power set, the structure measure theory deliberately avoids because countable additivity on all subsets is impossible. The Banach-Tarski paradox is exactly the statement that no finitely additive isometry-invariant measure on all subsets of $R^{3}$ extends Lebesgue measure, because $S O (3)$ contains $F_{2}$ ; the Vitali non-measurable set built there is the one-dimensional shadow of the same choice-driven phenomenon. Amenability of the isometry groups of $R^{1}$ and $R^{2}$ is what rescues a finitely additive invariant extension in low dimensions, where no paradox occurs.
Nuclearity and exactness of group C*-algebras 39.05.01. Theorem 8 makes the bridge to operator algebras: $C_{r}^{*} (G)$ is nuclear if and only if $G$ is amenable, and the Følner sequence is precisely the source of the completely positive finite-rank approximations witnessing the completely positive approximation property. This is the central insight that motivates placing amenable-group theory inside the nuclearity-exactness chapter, and it generalises from groups to amenable actions, where amenability of a group action yields nuclearity of the reduced crossed product.

Historical & philosophical context Master

The subject begins with John von Neumann's 1929 paper Zur allgemeinen Theorie des Maßes ^{[von Neumann 1929]}, written in response to the 1924 Banach-Tarski paradox. Von Neumann isolated the property — a left-invariant finitely additive measure on all subsets — that a group must lack for paradoxical decompositions to be possible, and identified the free group $F_{2}$ as the obstruction. He proved the class of groups carrying such a measure is closed under subgroups, quotients, extensions, and directed unions, the closure properties that still organise the theory. The term amenable (a pun on mean) was introduced later by Mahlon Day, who also formalised the fixed-point characterisation ^{[Day 1957]}.

Erling Følner's 1955 paper On groups with full Banach mean value ^{[Følner 1955]} supplied the combinatorial characterisation: a group has an invariant mean if and only if it contains finite sets that are asymptotically invariant under translation. Følner's condition turned an existential analytic statement into a constructive geometric one and became the working definition in geometric group theory and ergodic theory. Alfred Tarski had shown in 1929-1938 that the existence of an invariant measure is equivalent to the absence of a paradoxical decomposition ^{[Tarski 1938]}, giving the measure-theoretic dichotomy its sharp form.

Von Neumann's identification of $F_{2}$ as the obstruction hardened into the von Neumann conjecture: non-amenable if and only if containing $F_{2}$ . Sergei Adian's work on free Burnside groups in the 1970s and Alexander Ol'shanskii's 1980 construction of Tarski monsters ^{[Ol'shanskii 1980]} refuted it; Ol'shanskii and Sapir later produced a finitely presented counterexample. On the operator-algebra side, the equivalence of amenability with nuclearity of $C_{r}^{*} (G)$ and with injectivity of $L (G)$ — the latter via Connes' 1976 classification of injective factors — made amenability the organising hypothesis for finite-dimensional approximation, the theme Brown and Ozawa develop systematically ^{[Brown-Ozawa Ch. 2]}.

Bibliography Master

@article{vonNeumann1929,
  author  = {von Neumann, John},
  title   = {Zur allgemeinen Theorie des Ma\ss{}es},
  journal = {Fundamenta Mathematicae},
  volume  = {13},
  year    = {1929},
  pages   = {73--116}
}

@article{Folner1955,
  author  = {F{\o}lner, Erling},
  title   = {On groups with full Banach mean value},
  journal = {Mathematica Scandinavica},
  volume  = {3},
  year    = {1955},
  pages   = {243--254}
}

@article{Day1957,
  author  = {Day, Mahlon M.},
  title   = {Amenable semigroups},
  journal = {Illinois Journal of Mathematics},
  volume  = {1},
  year    = {1957},
  pages   = {509--544}
}

@article{Tarski1938,
  author  = {Tarski, Alfred},
  title   = {Algebraische Fassung des Ma\ss{}problems},
  journal = {Fundamenta Mathematicae},
  volume  = {31},
  year    = {1938},
  pages   = {47--66}
}

@article{Olshanskii1980,
  author  = {Ol'shanskii, Alexander Yu.},
  title   = {On the question of the existence of an invariant mean on a group},
  journal = {Uspekhi Matematicheskikh Nauk},
  volume  = {35},
  number  = {4},
  year    = {1980},
  pages   = {199--200}
}

@article{Connes1976,
  author  = {Connes, Alain},
  title   = {Classification of injective factors. Cases $II_1$, $II_\infty$, $III_\lambda$, $\lambda \neq 1$},
  journal = {Annals of Mathematics},
  volume  = {104},
  year    = {1976},
  pages   = {73--115}
}

@book{BrownOzawa2008,
  author    = {Brown, Nathanial P. and Ozawa, Narutaka},
  title     = {C*-Algebras and Finite-Dimensional Approximations},
  series    = {Graduate Studies in Mathematics},
  volume    = {88},
  publisher = {American Mathematical Society},
  year      = {2008}
}

@book{Runde2002,
  author    = {Runde, Volker},
  title     = {Lectures on Amenability},
  series    = {Lecture Notes in Mathematics},
  volume    = {1774},
  publisher = {Springer},
  year      = {2002}
}

@book{Wagon1985,
  author    = {Wagon, Stan},
  title     = {The Banach-Tarski Paradox},
  publisher = {Cambridge University Press},
  year      = {1985}
}

@article{AdianBurnside,
  author  = {Adian, Sergei I.},
  title   = {Random walks on free periodic groups},
  journal = {Izvestiya Akademii Nauk SSSR},
  volume  = {46},
  year    = {1982},
  pages   = {1139--1149}
}

Prerequisites

01.02.20
02.07.01

Tier anchors

beginner: averaging on groups: when can you take a fair average of an infinite collection?
intermediate: Brown-Ozawa, C*-Algebras and Finite-Dimensional Approximations §2.6; Runde, Lectures on Amenability §1
master: von Neumann 1929 Fund. Math. 13; Følner 1955 Math. Scand. 3; Day 1957 Illinois J. Math. 1; Ol'shanskii 1980; Brown-Ozawa Ch. 2; Wagon, The Banach-Tarski Paradox

References

Brown-Ozawa — C*-Algebras and Finite-Dimensional Approximations · Graduate Studies in Mathematics 88, AMS 2008, Ch. 2
Runde — Lectures on Amenability · Lecture Notes in Mathematics 1774, Springer 2002
von Neumann — Zur allgemeinen Theorie des Maßes · Fundamenta Mathematicae 13, 1929, 73-116
Følner — On groups with full Banach mean value · Mathematica Scandinavica 3, 1955, 243-254
Day — Amenable semigroups · Illinois Journal of Mathematics 1, 1957, 509-544
Ol'shanskii — On the question of the existence of an invariant mean on a group · Uspekhi Mat. Nauk 35, 1980, 199-200
Tarski — Algebraische Fassung des Maßproblems · Fundamenta Mathematicae 31, 1938, 47-66
Wagon — The Banach-Tarski Paradox · Cambridge University Press, 1985

Estimated time

beginner: 16m
intermediate: 45m
master: 80m