$\newcommand\Cb{C^\text b}$Let $\Cb(\mathbb R)$ be the C\*-algebra formed by all bounded, continuous, complex valued functions on $\mathbb R$.
Consider the action $\tau $ of $\mathbb R$ on $\Cb(\mathbb R)$ given by
$$
 \tau \_t(f)\mathclose|\_s = f(s-t), \quad \forall f\in \Cb(\mathbb R), \quad \forall s,t\in \mathbb R.
 $$


The spectrum of $\Cb(\mathbb R)$ is well known to be the Stone–Čech compactification $\beta (\mathbb R)$, so we
get an action
$\hat \tau $ of $\mathbb R$ on $\beta (\mathbb R)$
by duality, which clearly extends the usual action of $\mathbb R$ on itself by translation.


Evidently $\mathbb R$ is a $\hat \tau $-invariant open subset of $\beta (\mathbb R)$, whence the "corona"
$$
 \partial (\mathbb R)\mathrel{:=} \beta (\mathbb R)\setminus \mathbb R
 $$
is closed and
invariant.


It is easy to see that $\partial (\mathbb R)$ is not minimal among closed invariant subsets
because $\partial (\mathbb R)$ splits as the disjoint union of the following
two smaller closed invariant subsets:
$$
 \partial\_+ (\mathbb R) = \overline{(0, +\infty )} \setminus \mathbb R, \quad \text{and} \quad
 \partial\_- (\mathbb R) = \overline{(-\infty, 0)} \setminus \mathbb R.
 $$


**Question**. Are $\partial\_+ (\mathbb R)$ and $\partial\_- (\mathbb R)$ minimal? If not, what are examples of
minimal closed invariant subsets?



Minimal components of the translation action on the Stone–Čech compactification
$\newcommand{\Cb}{C^{\text b}}$I think I have a negative answer to my own question. Consider the function $f(x)=\sin(1/x)$, defined for $x$ in $(0,+\infty )$, and let us denote the graph of $f$ by $G$.


Let $\alpha $ be the arc length parametrization of $G$ oriented in such a way that $\alpha (s)$ approaches the vertical axis when $s\to +\infty $.


Next reparametrize $\alpha $ by defining $\gamma (s)=\alpha (\root 3 \of s)$, and observe that $$ \lim\_{s\to\infty }\|\gamma '(s)\| = 0. \tag 1 $$


Let us denote the closure of $G$ by $\mathcal R$ and let us consider the action $\sigma $ of $\mathbb R $ on $\mathcal R$, defined to be trivial on the vertical interval $I:= \{0\}\times [-1,1]$ and defined by $\sigma \_t(\gamma (s)) = \gamma (t+s)$ on $G$. It is evident that $\sigma $ is continuous on $\mathbb R\times G$, so let us prove that it is also continuous on $\mathbb R\times I$. We therefore suppose that $(t\_n,\gamma (s\_n))\to (t,x)$, with $x\in I$, and we must prove that $\gamma (t\_n+s\_n)\to x$. Observing that necessarily $s\_n\to+\infty $, this may be easily done employing the mean value Theorem and (1) as follows:
\begin{align}
\|\gamma (t\_n+s\_n)-x\|
& \leq \|\gamma (t\_n+s\_n)-\gamma (s\_n)\| + \|\gamma (s\_n)-x\| \cr
& \leq \sup\_{t\in [s\_n, s\_n+t\_n]}\|\gamma '(t)\| |t\_n| + \|\gamma (s\_n)-x\| \cr
& \to 0.
\end{align}


The map $\phi: \Cb(\mathcal R) \to \Cb(\mathbb R )$, defined by $\phi(f) = f\circ \gamma $, is clearly a covariant \*-monomorphism relative to $σ$ and $τ$. By duality it therefore leads to a covariant surjection $$ \hat \phi :\beta(\mathbb R) \to \widehat{\Cb(\mathcal R)}.$$ Picking any $x$ in the vertical interval $I$, hence a fixed point for $\sigma $, the evaluation at $x$ gives an element $\text{ev}\_x$ in $\widehat{\Cb(\mathcal R))}$ and $$ \hat\phi^{-1}(\{\text{ev}\_x\}) $$ is therefore a closed $\hat\tau$-invariant subset which may be proved to be a proper subset of $\partial \_+(\mathbb R)$ (as defined in the original post). Therefore $\partial \_+(\mathbb R)$ is not minimal!




---


The last part of the question, namely asking for examples of minimal closed invariant subsets is still wide open!


