Proof Clarification type decomposition of von Neumann algebras

Question

I'm reading a proof of the type decomposition theorem for von Neumann algebras (into a direct sum of 3 von Neumann algebras which are in turn of type 1,2,3)

One constructs maximal projections $p^1,p^2,p^3\in M$ so that $p^i$ is the largest central projection with $p^iM$ being of type $i$. These three projections are easily seen to be pairwise orthogonal. Now it remains to show that $1=p^1+p^2+p^3$. One does that by showing $q=1-p^1-p^2-p^3$ is the null projection.

The author goes on to show that $qM$ does not contain any non-zero finite projections(so $qM$ can't be of types 1,2) and that it can't be of type 3 either. Why does that imply that $q=0$ ?

I can't give a reference since the source is a lecture I attend.

Answer 1

I cannot comment without seeing your notes. One usually defines "type III" as "not type I nor II". This gives, by the maximality of $p^1$ and $p^2$, $p^3=1-p^1-p^2$. Because if $p^3$ is not $1$, then there would be a type I or II part, contradicting maximality.