Minimal projections on von Neumann Algebras

Question

A projection $p \neq 0$ in a von Neumann Algebra $A$ is called minimal, if for every projection $0\neq q\in A$ with $q \leq p$ already $q=p$.

I want to prove the following theorem:

For a minimal projection $p \in A$ the space $pAp$ is a division algebra.

Answer 1

Indeed $\mathbb{C}p=pAp$ which is isomormpic to the complex numbers.

Answer 2

Since $p$ is minimal, $pAp$ has no proper projections. By the spectral theorem, a von Neumann algebra is the norm-closure of the span of its projections. So $$ pAp=\text{span}\,\{0,p\}=\mathbb C p. $$