Let $G$ be an Hausdorff topological group and let $\beta G$ be the Stone-Čech compactification of $G$.
Now, $G$ is also a uniform space with respect to the so-called right uniformity. Let $S(G)$ be the Samuel compactification of $G$.
- Given a uniform space $( X , \mathcal{U} )$ there is a finest uniformity $\mathcal{U}_0$ on $X$ which is totally bounded and coarser than $\mathcal{U}$. The Samuel compactification of $( X , \mathcal{U} )$ is then the completion of $( X , \mathcal{U}_0 )$.
Are $\beta G$ and $S(G)$ the same space?
It seems the following.
A typical answer is negative, since the right-uniformity may be not fine and so there may exists a bounded continuous but not uniformly continuous function $f$ (for instance, put $G=\Bbb R$ endowed with the standard topology and $f(x)=\sin x^2$). Then $f$ extends to $\beta G$ but it does not extends to $S(G)$, because $f$ is not uniformly continuous with respect to $\mathcal U_0$ (because $\mathcal U_0\subset\mathcal U$).