  Given an action of a discrete countable group $G$ on a countable set
$\mathfrak{X}$, it is studied the relationship between properties of the
associated Calkin representation and the dynamics of the group action on the
boundary of the Stone-\v{C}ech compactification of $\mathfrak{X}$. The first
section contains results about amenability properties of actions of discrete
countable groups on non-separable spaces and is of independent interest. In the
second section these results are applied in order to translate regularity
properties of the Calkin representation and the topological amenability on the
Stone-{\v C}ech boundary within the common framework of measurable dynamics on
certain extensions of the Stone-{\v C}ech boundary of $\mathfrak{X}$.
