# comparison of two projections in a non-factor von Neumann algebra

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?

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.
• I suppose this can happen for small algebras like $\mathbb C \oplus \mathbb C$. What you are looking for is the comparison theorem. For a von Neumann algebra $M$ and projections $p,q$ in $M$ there exists a central projection $z$ such that $pz \preceq qz$ and $q(1-z) \preceq p(1-z)$. A nice example to think about is the von Neumann algebra of measurable essentially bounded functions from the interval $[0,1]$ to $M_2(\mathbb C)$. Oct 29 '20 at 16:51