In a factor $M$, we know that for any two projections $P$ and $Q$ in $M$, either $P\preceq Q$ or $Q\preceq P$ holds true. Here $\preceq$ denotes the Murray-von Neumann subequivalence of two projections. Is there a von Neumann algebra which is not a factor, where such comparison holds true for any two projections? If yes, is there a characterization for such algebras?



comparison of two projections in a non-factor von Neumann algebra
This can never happen. Let $M$ be a von Neumann algebra with a nontrivial center $Z(M)$. Take two nonzero mutually orthogonal projections $p,q \in Z(M)$. Suppose these projections are comparable. Then w.l.o.g. we have a partial isometry $v \in M$ so that $vv^\*=p$ and $v^\*v \leq q$ is a projection. Since $p$ is a central projection and $pq=0$,
$$0 \neq v^\*v=v^\*vv^\*v$$
$$=vv^\*v^\*v=pv^\*v=0.$$
Some intuition for von Neumann algebras with nontrivial centers can be gained from disintegration results. In the separable case any von Neumann algebra is the direct integral of factors over a compact Hausdorff space. A proof of this can be found in Dixmier's "von Neumann Algebras" or in Takesaki.


