Proof. (1): Suppose that $M$ has a minimal projection. By Theorem 43.9.15, there exists a complex Hilbert space $H$ such that $M \iso B(H)$. Let $\seqi{e}\subset H$ be an orthonormal basis, then $e_{i} \otimes e_{i} \in \text{Proj}(M)$ is minimal for all $i \in I$, and all such rank one projections are equivalent. As such, for any $j \in I$,

so $I$ must be finite, and $M \iso M_{|I|}(\complex)$.

(2): Suppose that $M$ has no minimal projections. By Lemma 43.9.13, $M$ has no abelian projections. As $\tau$ is a faithful tracial state, $1_{M}$ is finite. Therefore $M$ is a $\vnII_{1}$ factor.$\square$