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:



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> **Question** Is it true that $\pi$ maps clopen sets into clopen sets?
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Continuous surjection between spectra of commutative von Neumann algebras

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> Is it true that π maps clopen sets into clopen sets?
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This is true if and only if the inclusion $V\_1→V\_2$ is a morphism of von Neumann algebras, i.e., its image is closed in the ultraweak topology.


See Proposition 2.48 in [arXiv:2005.05284](https://arxiv.org/abs/2005.05284).


