On Fixed Points of L\"{u}ders Operation

In this paper, we prove that if $\mathcal{A}=\{E_i\}_{i=1}^{n}$ is a finite commutative quantum measurement, then the fixed points set of L\"{u}ders operation $L_{{\cal A}}$ is the commutant ${\cal A}'$ of ${\cal A}$, the result answers an open problem partially. We also give a concrete example of a L\"{u}ders operation $L_{{\cal A}}$ with $n=3$ such that $L_{{\cal A}}(B)=B$ does not imply that the quantum effect $B$ commutes with all $E_1, E_2$ and $E_3$, this example answers another open problem.