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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?

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    $\begingroup$ might be of use: a subset $S\subset X_i$ is clopen iff $S=\{\gamma\in X_i: \gamma(p)=0\}$ is the hull of some projection $p\in V_i$. $\endgroup$
    – Onur Oktay
    Aug 12, 2022 at 13:48
  • $\begingroup$ Yes, but does it help? If I have a projection $P$ in $V_2$ it is of the form $\chi_W$ for some clopen $W$. Then of course I can consider a function $\chi_{\pi[W]}$ but unless I know that $\pi$ preserves clopen sets, I don;t know whether this function in fact lies in $V_1$. Note that there is no natural map $V_2 \to V_1$. $\endgroup$
    – truebaran
    Aug 12, 2022 at 15:41
  • $\begingroup$ commutative VN-algebras are injective C*-algebras. Thus, there exists a conditional expectation $V_2\to V_1$. Also, commutative VN-algebras are isometrically injective Banach spaces, so $V_1$ is 1-complemented in $V_2$ as a Banach space. $\endgroup$
    – Onur Oktay
    Aug 12, 2022 at 19:11

1 Answer 1

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Is it true that π maps clopen sets into clopen sets?

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.

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  • $\begingroup$ Thank you for your answer-also thank you for the very informative paper! $\endgroup$
    – truebaran
    Aug 19, 2022 at 13:33

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