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Let $X$ be an extremally disconnected (i.e. such that the closure of open sets is open) compact Hausdorff space. Then

$*_1$ $C(X)$ is the space of continuous functions $f: X \to \mathbb{C}$,

$*_2$ $C^+(X)$ is the space of continuous functions $f: X \to \mathbb{S}^2 = \mathbb{C} \cup \{\infty\}$ such that the pre-image of $\infty$ is nowhere dense ($\mathbb{S}^2$ is the one point compactification of $\mathbb{C}$).

Suppose $\mathbb{B}$ is a complete Boolean algebra, and consider the Boolean valued universe $V^{\mathbb{B}}$. Let $St(\mathbb{B})$ be the Stone space of $\mathbb{B}$. Then there is a correspondence between:

$\star_1$ the family of $\mathbb{B}$-names for complex numbers in the boolean valued model $V^{\mathbb{B}}$: $\dot{\mathbb{B}} = \{\tau \in V^{\mathbb{B}}: \parallel \tau \text{~is a complex number~}\parallel_{\mathbb{B}} = 1_{\mathbb{B}} \}$

and

$\star_2$ $C^+(St(\mathbb{B}))$.

Remark. Let $\mathbb{B}$ be the complete boolean algebra given by Lebesgue measurable sets modulo Lebesgue null sets. Then $C(St(\mathbb{B}))$ is isomorphic to $L^\infty(\mathbb{R})$.

For more details, see [1], [2] below.

These results, motivate us to ask the following question:

Question. Suppose $\mathbb{B}$ is a complete Boolean algebra, or even a simple one, say the complete boolean algebra given by Lebesgue measurable sets modulo Lebesgue null sets. Is there a statement $RH(\mathbb{B})$ such that:

$V^{\mathbb{B}}\models $the Riemann hypothesis holds $\iff~ RH(\mathbb{B})$ holds for $C^+(St(\mathbb{B}))$.

Remark As the Riemann hypothesis is a $\Pi_1$ statement, by the Shoenfield absoluteness theorem, its truth does not change by forcing. Thus for any complete Boolean algebra $\mathbb{B}$, we have $V^{\mathbb{B}}\models $``the Riemann hypothesis holds'' iff the Riemann hypothesis holds.

[1] Vaccaro, Andrea; Viale, Matteo; Generic absoluteness and boolean names for elements of a Polish space, Boll. Unione Mat. Ital. 10 (2017), no. 3, 293–319.

[2] Viale, Matteo; Forcing the truth of a weak form of Schanuel's conjecture, Confluentes Mathematici. 8 (2016), no. 2, 59–83.

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  • $\begingroup$ Isn't "$V^{\mathbb{B}}\models$ the Riemann hypothesis holds" such a statement? Or are you looking for some "functional analytic" statement about the space $C^+(St(\mathbb{B}))$? $\endgroup$
    – new account
    Commented Jun 27 at 6:08
  • $\begingroup$ How is it possible $f$ take value $\infty$ .?You assumed that $X$ is compact so $f(X)$ is far from the north pole, $\endgroup$
    – Ali Taghavi
    Commented Jun 27 at 18:14
  • $\begingroup$ I am talking about $C^+(X)$ definition $\endgroup$
    – Ali Taghavi
    Commented Jun 27 at 18:19
  • $\begingroup$ Moreover since the algebra $C(X)$ is a commutative model of a von Neumann algebra, I googled "Von neumann algebra + RH" I got some things relevant $\endgroup$
    – Ali Taghavi
    Commented Jun 27 at 20:55
  • $\begingroup$ I am realy curious abiut the reason you impose the condition "The preimage of $\infty$ is nowhere dense" when we are sure that the range of $f$ does not approach infinity $\endgroup$
    – Ali Taghavi
    Commented Jun 28 at 1:52

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