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 $q=1/2\begin{pmatrix} 1+z& x+yi\\ x-yi& 1-z \end{pmatrix}$?


It can be easily shown that every two homotopic projections $p,q$ have  a  bisector  projection. Here is  a proof: 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 $$\phi (t)= |p\gamma(t)-\gamma(t)|-|q\gamma (t)-\gamma(t)|$$ In fact we have $\phi(0)\phi(1)<0$. For if  $pq=q$ then $qp=q$ then $e=p-q$ would  be  an idempotent with $e^*=-e$, a  contradiction.