# Equivalence of projections in a von Neumann algebra

Let $M\subset B(H)$ be a von Neumann algebra. Also $p$ and $q$ are projections in $M$ such that $pMp$ is star isomorphic to $qMq$. Does this imply that $p$ and $q$ are equivalent in the Murray-von Neumann sense?

• Do you require $M$ to be a factor? In any case there are counterexamples. For instance take $M = R$ the hyperfinite factor. Any $p R p$ gives another hyperfinite factor. Commented Jul 19, 2018 at 14:44
• The notion you are after is probably that of "fundamental group" Commented Jul 19, 2018 at 14:44

No. Take $$M=\mathbb C^2\subset M_2(\mathbb C)$$, $$p=E_{11}$$, $$q=E_{22}$$. Then $$pMp=\mathbb C\oplus 0\simeq 0\oplus\mathbb C=qMq,$$ but $$p$$ and $$q$$ are not equivalent (equivalence in a commutative algebra is equality).
Even with factors, the assertion is not true. If $$M$$ is the hyperfinite II$$_1$$-factor and $$p\in M$$ is any projection (alternatively, $$M$$ is any II$$_1$$-factor and $$p$$ is in the fundamental group) then $$M\simeq pMp$$ while $$p$$ is not equivalent to the identity.