Spectrum of Laplace operator $\Delta\colon C^{\infty}(\Omega)\to C^{\infty}(\Omega)$.

Question

Let $\Omega\subset\mathbb{R}^{n}$ be a compact set and let $$\Delta\colon C^{\infty}(\Omega)\to C^{\infty}(\Omega)$$ be the Laplace operator defined on $C^{\infty}(\Omega)$. I have some questions about this operator:

1. Is there a norm on $C^{\infty}(\Omega)$ that makes $B(C^{\infty}(\Omega))$ (= set consisting of bounded linear operators $C^{\infty}(\Omega)\to C^{\infty}(\Omega)$) into a Banach Algebra. And do we then have $\Delta\in B(C^{\infty}(\Omega))$?
2. What can we say about the spectrum of $\Delta$? (Some sources also say that we need boundary conditions, why?)
3. Why do people consider the completion of $C^{\infty}(\Omega)$ and extend the definition of $\Delta$ to this completion?