When set of projections is closed under multiplication Let $\mathcal A\subset B(H)$ be an unital $C^*$ algebra of operators on a Hilbert space $H$. Let's denote by $\mathcal P$ the set of projections in $\mathcal A$, that is $\mathcal P:=\{a\in \mathcal A \; | \; a=a^2=a^*\}$. 
I'm wondering when $\mathcal P$ is closed under composition of operators. I was able to prove the following:
$\big(\mathcal P \text{ is closed under multplication}\big)\Leftrightarrow \big(\forall_{a,b\in \mathcal P}\; ab=ba\big)$
If $\mathcal A$ is a von Neumann algebra then projections spans set which is dense in $\mathcal A$ so in this case $\big(\mathcal P \text{ is closed under multiplication}\big)\Leftrightarrow \big(\mathcal A \text{ is abelian}\big)$. Can something similar be proven in general(without assuming $\mathcal A$ being a von Neumann algebra)?.

No. If $\mathcal A=C_r(\mathbb F_2)$ (or any other projectionless unital C$^*$-algebra), then $$\mathcal P=\{0,I\}$$ is closed under multiplication. 
