There is an adjunction $\text{Bool}^{op} \leftrightarrow \text{Set}$ between boolean algebras and sets which sends a boolean algebra to the set of its prime ideals and a set $X$ to $[X, \mathbb{F}\_2]\_{\text{Set}}$. This adjunction factors as $\text{Bool}^{op} \cong \text{PF} \leftrightarrow \text{Set}$, where $\text{PF}$ is the category of profinite sets.


there is an adjunction $C^\* \text{-alg}^{op} \leftrightarrow \text{Top}$ between commutative unitial $C^\*$-algebras and top which sends a commutative unital $C^\*$-algebra to the set of its prime ideals (edit: this should be maximal ideals, equipped with the hull-kernel topology) and a set $X$ to $C^\*(X, \mathbb{R})$. This adjunction factors as $C^\* \text{-alg}^{op} \cong \text{CH} \leftrightarrow \text{Top}$, where $\text{CH}$ is the category of compact hausdorff spaces.


These two adjunctions seem related in some sense. My question is whether there is an analogue of the following extension, for compact hausdorff spaces instead of profinite sets:



> 
> **Theorem:** Define a functor $R \text{-alg} \rightarrow \text{Bool}$ sending an $R$-algebra to the boolean algebra given by the set of its idempotents (see Pierce spectrum functor). This functor is right adjoint. So we have an adjunction $R \text{-alg} \leftrightarrow \text{Set}$.
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Now I am looking for a functor $\text{top-} R \text{-alg} \rightarrow C^\*\text{-alg}$ or maybe a functor $\text{top-} R \text{-alg} \rightarrow \text{CH}$, or some tweaking of this, which would be analogous to the above theorem.



Functor from rings into compact Hausdorff spaces
Here is an answer to your question about monadicity, as it's too long for a comment. I will not fill in every detail, so if you follow along there will be several definitions that need to be expanded and diagrams that need to be shown to commute in order to verify everything, so let me know in a comment if you get seriously stuck.


In short, the first adjunction is not monadic, but the second one is.


For the first adjunction, the left adjoint is (or is isomorphic to, the way it's set up in the question) the contravariant power set functor $\newcommand{\powerset}{\mathcal{P}}\powerset : \mathbf{Set} \rightarrow \mathbf{Bool}^{\mathrm{op}}$, and the right adjoint $\Sigma : \mathbf{Bool}^{\mathrm{op}} \rightarrow \mathbf{Set}$ takes a Boolean algebra $A$ to the underlying set of its Stone space $\Sigma(A)$. Then the monad is $\newcommand{\U}{\mathcal{U}}\U = \Sigma \powerset$, which is known as the *ultrafilter monad*. Manes proved that the Eilenberg-Moore category $\newcommand{\EM}{\mathcal{EM}}\EM(\U)$ is equivalent to the category of compact Hausdorff spaces (by showing that it arises from a *different* adjunction between $\mathbf{Set}$ and the category of compact Hausdorff spaces, see *e.g.* VI.9 in Mac Lane's *Categories for the Working Mathematician*). From this, we can see that the comparison functor $\mathbf{Bool}^{\mathrm{op}} \rightarrow \EM(\U)$ is not essentially surjective, because it lands in Stone spaces, and $[0,1]$ is a compact Hausdorff space that is not homeomorphic to a Stone space. Therefore the $(\powerset,\Sigma)$ adjunction is not monadic.


The second adjunction is not stated correctly in the question, so I'll give the correct version of it. The functor $\newcommand{\Cstar}{\mathbf{C^\*\mbox{-}alg}}\newcommand{\Top}{\mathbf{Top}}\newcommand{\CH}{\mathbf{CH}}\newcommand{\op}{^{\mathrm{op}}}\Sigma : \Cstar\op \rightarrow \Top$ takes a C$^\*$-algebra $A$ to its Gel'fand spectrum $\Sigma(A)$, which is the set of *maximal* ideals equipped with the hull-kernel topology (the set of prime ideals is very [awkward to handle](https://mathoverflow.net/questions/35793/prime-ideals-in-c0-1) even for $C([0,1])$ and doesn't work for a duality in this case). The functor $C\_b : \Top \rightarrow \Cstar\op$ takes a space $X$ to the C$^\*$-algebra of *bounded* complex-valued functions (this can be done with real-valued functions, as in Johnstone's book, but I prefer not to do it that way for compatibility with the noncommutative case. For a general topological space $X$, boundedness must be required, and the unbounded functions $C(X)$ do not form a C$^\*$-algebra.) This time, the monad $\Sigma \circ C\_b : \Top \rightarrow \Top$ is a form of the Stone-Čech compactification (this can be proved by verifying the universal property). So we write $\beta = \Sigma \circ C\_b$ for short.


Now, we can prove that the $(C\_b,\Sigma)$ adjunction is monadic as follows. If $(X,\alpha\_X : \beta(X) \rightarrow X)$ is an Eilenberg-Moore algebra of $\beta$, then we have $\alpha\_X \circ \eta\_X = \mathrm{id}\_X$, where $\eta\_X$ is the unit. This implies that $\alpha\_X$ is surjective, which, since $\beta(X)$ is compact, implies that $X$ is compact. It also implies that $\eta\_X$ is injective, which, since $\beta(X)$ is Hausdorff, implies that $X$ is Hausdorff. Therefore $X$ is a compact Hausdorff space, $\eta\_X$ is an isomorphism, and so $\alpha\_X$ is required to be its inverse. This means $\EM(\beta) \simeq \CH$, and by Gel'fand duality the comparison functor $\Cstar\op \rightarrow \CH \simeq \EM(\beta)$ is an equivalence of categories, *i.e.* the $(C\_b,\Sigma)$ adjunction is monadic.


