Canonical action of a countable discrete group $G$ on its stone-cech compactification $\beta G$

Question

When I read some materials in topological dynamics, I met words: "canonical action of a countable discrete group $G$ on its stone-cech compactification $\beta G$" without any definition. I know that $\beta G$ can be defined via ultrafilters and the topology is the Stone topology.

Could you help me to figure out the definition of the ''canonical'' action? I cannot find one.

Thank you for all helps!

Answer 1

Let $g \in G$. Then the translation $x \to gx$ defines an action from $G$ onto itself. This action extends to a continuous action from $\beta G$ to $\beta G$.

Answer 2

I tend to think of $\beta G$ differently. Say $B(G)$ is the space of bounded (continuous) complex-valued functions on $G$. Then $B(G)$ is a Banach algebra, and $\beta G$ is the maximal ideal space of $B(G)$, which is to say the space of complex homomorphisms of $B(G)$.

Say $\phi\in\beta G$, which in our current formulation says $\phi:B(G)\to\Bbb C$ is a homomorphism. If $g\in G$ then $$f\mapsto\phi(f\circ g^{-1})$$is another complex homomorphism of $B(G)$. So we can define $$T_g:\beta G\to\beta G$$by $$(T_g\phi)(f)=\phi(f\circ g^{-1}).$$

Answer 3

Let's say the translate of a set by a group element is the set obtained by translating each element of the set individually. An ultrafilter is a collection of sets with certain properties. An element of the group acts on an ultrafilter by translating the corresponding sets individually, giving rise to a new ultrafilter.