Suppose that $V_1,V_2$ are two commutative von Neumann algebras and $V_1 \subset V_2$. Being in particular commutative $C^*$-algebras we have that $V_1 \cong C(X_1), V_2 \cong C(X_2)$ for some topological compact spaces $X_1,X_2$. Our inclusion $V_1 \subset V_2$ yields a continuous surjection $\pi:X_2 \to X_1$ going in the opposite direction. Since $V_1,V_2$ are von Neumann algebras then $X_1,X_2$ are in fact very special (the so called hyperstonean spaces). I wonder whether in this case:
Question Is it true that $\pi$ maps clopen sets into clopen sets?