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.

Solution 1:

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.