I am having some trouble constructing the Stone-Čech compactification of a locally compact Hausdorff space $X$ using theory of $C^*$-algebras. I did some search but could not find a good answer on this.
Let's focus on the case $X=\mathbb{R}$. The space of bounded complex-valued functions $C_b(\mathbb{R})$ is a commutative unital $C^*$-algebra hence $C_b(\mathbb{R})\cong C(\mathcal{M})$, where $\mathcal{M}$ is the maximal ideal space, which is compact and Hausdorff.
It should be the case that $\mathcal{M}\cong\beta\mathbb{R}$, and it is not difficult to show that by identify $t\in\mathbb{R}$ with the evaluation at $t$, we have a homeomorphism between $\mathbb{R}$ and a subspace of $\mathcal{M}$.
But we still need to show this subspace is dense in $\mathcal{M}$. This is where I am having troubles (and I guess this is the whole point of the proof).
Can someone give a hint? Thanks!