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Let us consider two projections $e_1$ and $e_2$ on the Hilbert space $H$ with $e_1e_2=0$. Does there exists any projection $p$ with

$$\overline{pe_1(H)}=\overline{pe_2(H)}=\overline{p(e_1+e_2)(H)}\neq0$$

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Wlog we can assume that $$e_1=\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix} \mbox{ and } e_2=\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix}$$ with respect to a certain orthogonal decomposition of $H=H_1\oplus H_2\oplus H_3$.

Then it suffices to set $$p=\begin{pmatrix}1/2&1/2&0\\1/2&1/2&0\\0&0&0\end{pmatrix}.$$

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