contruced a $*$-isomorphism between two von neumann algebras

Question

Let $M$ be a von Neumann algebra and $p$ is a projection in $M$. Can we contruct a $*$-isomorphism between $M$ and $pMp$?

Answer 1

Try with $M=\mathbb C\oplus\mathbb C$ and $p=(1,0)$.

The question is more relevant in the case of II$_1$-factors, and it was been studied already by Murray and von Neumann 80 years ago (the Fundamental Group). From the uniqueness of the hyperfinite II$_1$-factor $R$ it follows that $R\simeq pRp$ for all nonzero projections $p\in R$. At the other end of the spectrum, Popa has constructed a II$_1$-factor $M$ such that $M\not\simeq pMp$ for all nontrivial projection $p\in M$.