# Equivalent definitions of type II von Neumann algebras?

I’ve seen the following two definitions for type II von Neumann algebras and I am wondering if they are equivalent. The first is that a von Neumann algebra $$M$$ is type II if it has no non-zero abelian projections but has a finite projection with central carrier $$I$$, and the second is that a von Neumann algebra $$M$$ is type II if it has no non-zero abelian projections and $$I$$ is semi-finite (i.e., $$I=\sum P_{i}$$ where $$\{P_{i}\}$$ is an orthogonal family of finite projections). I do not see how the second could imply the first, for example, unless we can assume all the $$P_{i}$$ are equivalent, but I don’t know why that would be possible.

The projections $$P_j$$ don't have to be equivalent, even if $$M$$ is a factor. And, in general, they will not be comparable if they live in different central summands.
Suppose that $$M$$ has no non-zero abelian projection and $$I=\sum_jP_j$$ with each $$P_j$$ finite. Since $$M$$ has no non-zero abelian projections, by the Type Decomposition it is of the form $$M_2\oplus M_3$$, with $$M_2$$ of type II and $$M_3$$ of type III. These are given by two central projections $$Q_2$$ and $$Q_3$$ with $$Q_2+Q_3=I$$. We have $$Q_3=\sum_jQ_3P_j$$, and each $$Q_3P_j\in M_3$$ is finite, since $$Q_3P_j\leq P_j$$. Being type III, $$M_3$$ has no nonzero finite projections, so $$Q_3P_j=0$$. But then $$Q_3=\sum_jQ_3P_j=0$$, and $$M=M_2$$ is type II.