Prove a von Neumann algebra is diffuse

Question

In Lemma 2.2, The authors want to prove $M_{\varphi}$ is diffuse. How to check that $eMe$ falls in $A$?

Answer 1

I don't this it's automatically obvious that "$eMe\subset A$". But it is of course true, and it follows easily from an attempt to see that if $e$ is minimal in $A$ then it is minimal in $M$.

Indeed, let $f\in eMe$ be a projection. For any $a\in A$, and using that $Ae=\mathbb C\,e$, $$ fa=fea=f\lambda_ae=\lambda_aef=aef=af. $$ So $f\in A'=A$. Then $f\in Ae=\mathbb Ce$. Thus $eMe=\mathbb C\,e$.