Let $A$ be a C\*-algebra and let $\mathcal{P}(A)$ denote the set of projections in $A$. If $p\in\mathcal{P}(A)$ commutes with everything in $\mathcal{P}(A)$ does it necessarily commute with everything in $A$? I feel like I should/did know this but I'm currently drawing a blank, so any help would be much appreciated.



Can non-central projections still commute with all other projections?
The answer is yes, but I don't know where it is written. If $p$ is not central, then $pAp^\perp\neq\{0\}$ and one can take $x\in pAp^\perp$ such that $0<\|x\|<1/2$. Then, $$q:=\left(\begin{array}{cc} \frac{1+\sqrt{1-4xx^\*}}{2} & x\\ x^\*& \frac{1-\sqrt{1-4x^\*x}}{2}\end{array}\right)\quad{\rm in}\quad \left(\begin{array}{cc} pAp & pAp^\perp \\ p^\perp Ap & p^\perp A p^\perp\end{array}\right),$$ or more formally, $$q=\frac{p+\sqrt{p-4xx^\*}}{2}+x+x^\*+\frac{p^\perp-\sqrt{p^\perp-4x^\*x}}{2}\in A$$ is a self-adjoint projection which doesn't commute with $p$.
Here $p^\perp=1-p$ belongs to the multiplier of $A$. Even if $A$ is not unital, $p^\perp A p^\perp$ makes sense as a hereditary subalgebra of $A$ and the functional calculus also makes sense in the non-unital $\mathrm{C}^\*$-algebra $p^\perp A p^\perp$, because the function $f(t)=(1-(1-4t)^{1/2})/2$ vanishes at $t=0$.


