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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?

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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.

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  • $\begingroup$ One more question: Can two non-central projections always be compared in some non-factor von Neumann algebras? $\endgroup$ Oct 29 '20 at 6:47
  • $\begingroup$ 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 $M_2(\mathbb C)$. $\endgroup$ Oct 29 '20 at 16:51

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