Let  $p,q$  be  two  projections of a  $C^*$  algebra. A projection $l$ is  called a  bisector projection to $(p,q)$ if  $$|pl-l|=|ql-l|$$ The motivation comes from the geometric  intuition of "Bisector" in plane  geometry.


>Assume  that  two projections $p,q$ are  similar or  Mourray von Neumann equivalent. Does they  admit  a bisector  projection? What is the answer to this question for the particular case $A=M_2(C(S^2))$ with the projections $p=\begin{pmatrix} 1&0\\0&0\end{pmatrix}$ and $q=1/2\begin{pmatrix} 1+z& x+yi\\ x-yi& 1-z \end{pmatrix}$?


**Remark:** One can prove that for two homotopic projections $p,q$ there always exist a bisector projection $l$. To prove this we may assume that $pq\neq q$ otherwise $l=1-p$ is a bisector projection to $p,q$.  Let  $\gamma (t)$  be a  curve of  projections with  $\gamma(0)=p,\; \gamma(1)=q$. Now  we apply the  intermediate value theorem to the  continuous  function $$\phi (t)= |p\gamma(t)-\gamma(t)|-|q\gamma (t)-\gamma(t)|$$ Observe that $\phi(0)\phi(1)<0$.  This completes the proof.