Let $\beta \mathbb{R}$ be the Stone-Čech compactification of $\mathbb{R}$ (with euclidean topology) and $C_0(\mathbb{R})$ the $C^*$-algebra of continuous complex-valued functions vanishing at infinity. Why is $C(\beta \mathbb{R})/C_0(\mathbb{R})\cong C(\beta \mathbb{R}\setminus \mathbb{R})$ as $C^*$-algebras?
My problem is that I'm stuck to understand the Stone-Čech compactification of $\mathbb{R}$ and I dont know how to understand $\beta \mathbb{R}\setminus \mathbb{R}$. I know that one can identify $C(\beta \mathbb{R})$ with $C_b(\mathbb{R})$ as $C^*$-algebras, the continuous bounded complex-valued functions on $\mathbb{R}$.
You have $$C_0(\mathbb R)=\{f\in C(\beta\mathbb R):\ f|_{\beta\mathbb R\setminus\mathbb R}=0\}.$$