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 $[\mathbb{F}_2 : \alpha(\mathbb{F}_2)]$ is infinite as the family $((ab)^k)_{k \geq 1}$ yields mutually distinct left-cosets in $\mathbb{F}_2 / \alpha(\mathbb{F}_2)$.

Let $D := C^*(\{ (e_{w,n})_{w \in \mathbb{F}_2,n \in \mathbb{N}} \mid e_{w,n} \text{ is a projection and } \mathcal{R} \text{ holds.}\})$, where relation $\mathcal{R}$ says that, for $m<n,v,w \in \mathbb{F}_2$, $$e_{v,m} e_{w,n} = \begin{cases} e_{vw',n} & \text{if } v^{-1}w \in \alpha^m(\mathbb{F}_2)\alpha^n(\mathbb{F}_2) = \alpha^m(\mathbb{F}_2), \\ 0 & \text{else,}\end{cases}$$ where $w'$ satisfies $vw' \in w\alpha^n(\mathbb{F}_2)$. $w'$ is uniquely determined up to right multiplication by some element from $\alpha^n(\mathbb{F}_2)$. In particular, the projections commute and $e_{v,n} = e_{w,n}$ is equivalent to $v^{-1}w \in \alpha^n(\mathbb{F}_2)$. Now $\mathbb{F}_2$ acts on $D$ simply by translation on the first index, i.e. $\tau_v(e_{w,n}) = e_{vw,n}$.

$\textbf{Question :}$ Is $\mathbb{F}_2 \stackrel{\tau}{\curvearrowright}D$ amenable?

The following additional information may be helpful: We can ask the equivalent question for the corresponding action $\mathbb{F}_2 \stackrel{\hat{\tau}}{\curvearrowright}\hat{D}$ on the spectrum $\hat{D}$ of $D$. Since $D$ is unital and generated by commuting projections, $\hat{D}$ is a totally disconnected, compact Hausdorff space. If we let $D_n := C^*(\{e_{w,m} \mid w \in \mathbb{F}_2, 0 \leq m \leq n \}) \subset D$, then $D = \overline{\bigcup_{n \in \mathbb{N}} D_n} = \varinjlim D_n$, so $\hat{D} = \varprojlim \hat{D}_n$. From this one sees that a basis for the topology on $\hat{D}$ is given by cylinder sets of the form $$C_{(v_0,n_0),(v_1,n_1),\dots,(v_k,n_k)} = \{ \chi \in \hat{D} \mid \chi(e_{v_0,n_0}) = 1, \chi(e_{v_i,n_i}) = 0 \text{ for all } 1 \leq i \leq k\}$$ with $v_i \in \mathbb{F}_2$ and $n_i \in \mathbb{N}$.

Let us look at the map $\iota:\mathbb{F}_2 \longrightarrow \hat{D}$ given by $$\iota(w)(e_{v,n}) = \begin{cases} 1 & \text{if } v^{-1}w \in \alpha^n(\mathbb{F}_2), \\ 0 & \text{else.}\end{cases}$$ Using the basis for the topology from before, it is apparent that $\iota$ has dense image. Since $\bigcap_{n \in \mathbb{N}} \alpha^n(\mathbb{F}_2) = \{1_{\mathbb{F}_2}\}$, this is actually a dense embedding (but note that $\hat{D}$ is not a group). From this, on sees that $\hat{\tau}$ is topologically free and minimal since it acts transitively on $\iota(\mathbb{F}_2)$.

**A short comment:** This exposition can be thought of as a minimal example for dynamical systems with a free group in countably many generators and a bunch of suitable group endomorphisms. Understanding amenability of the corresponding $\tau$ for such dynamical systems would make me really happy - but my current knowledge on techniques to show amenability of actions for highly non-amenable groups is fairly limited. So any hints on what may yield insights is appreciated.

**Concrete picture for $D$:** From the initial data, we can pass to the semidirect product $S := \mathbb{F}_2 \rtimes_\alpha \mathbb{N}$. On the Hilbert space $\ell^2(S)$, let $E_{w,n} \in \mathcal{L}(\ell^2(S))$ be the orthogonal projection onto the right ideal $$(w,n)S = \{(v,m) \mid v \in w\alpha^n(\mathbb{F}_2), m \geq n\} \subset S.$$
Then $e_{w,n} \mapsto E_{w,n}$ defines an isomorphism $D \longrightarrow C^*((E_{w,n})_{w \in \mathbb{F}_2,n \in \mathbb{N}})$. This uses the idea of independent constructible right ideals for semigroup C*-algebras developed by Xin Li in http://arxiv.org/abs/1203.0021 (Def 2.5).