In pages 5-6 of the article https://arxiv.org/pdf/quant-ph/9911020.pdf, the notion of a spectral presheaf is basically introduced as (in a more contemporany notation):
Definition(Valuation): A functional $v:\mathfrak{V}_{sa} \rightarrow \mathbb{R}$, form the self-adjoint elements of a $C^*$-algebra $\mathfrak{V}$ is called a valuation if for each $A \in \mathfrak{V}_{sa}$, $\,\,v(A) \in \sigma(A)$ and $v(f(A))=f(v(A))$ for each continuous function $f:\sigma(A) \rightarrow \mathbb{R}$, where $f(A)$ is the continuous functional calculus of $A$ through $f$.
Definition(Gel'fand spectrum): Let $\mathfrak{V} \subseteq \mathfrak{M}$ be an unital abelian sub-$*$-algebra of the von Neumann algebra $\mathfrak{M}$. We denote by $\Sigma_\mathfrak{V}$ the so called Gel'fand spectrum of $\mathfrak{V}$, the set composed of positive linear functionals, e.g. $\lambda:\mathfrak{V} \rightarrow \mathbb{C}$, these functionals are called characters and are of norm $1$, i.e. $\|\lambda\|_{\mathfrak{V}'} =1$, they are also $*$-algebra homomorphisms, that is: $$\forall \, A,B \in \mathfrak{V},\,\,\,\,\,\, \lambda(AB^*) = \lambda(A)\overline{\lambda(B)},\,\,\,\,\,\,\,\,\,\,\,\, \forall \, \lambda \in \Xi_\mathfrak{V}. $$
Definition(Context and generated context): An unital abelian sub-$*$-algebra $\mathfrak{V}$ that agrees on the unit from a unital $*$-algebra $\mathfrak{M}$ ($\mathfrak{V}\subseteq \mathfrak{M}$) is called a context. Given a $\{A_n\}_{n∈I}$ a collection of self-adjoint commuting elements, then there is at least one context containing $\{A_n\}_{n∈I}$(?!?), then we can consider the smallest of all such contexts, and this context is said to be the context generated by $\{A_n\}_{n∈I}$, denoted by $\mathcal{C}(\{A_n\}_{n∈I})$. The collection of all contexts of $\mathfrak{M}$ is denoted by $\mathfrak{C}(\mathfrak{M})$.
Definition (Spectral Presheaf): For a von Neumann algebra $\mathfrak{M}$, we define the Spectral presheaf $\underline{\,\Sigma\,}$ as the contravariant functor $\underline{\,\Sigma\,}: \mathfrak{C}(\mathfrak{M}) \rightarrow \textbf{Set}$, that maps each context $V$ (object) in $\mathfrak{C}(\mathfrak{M})$ to $\underline{\,\Sigma\,}(V) \equiv \underline{\,\Sigma\,}_V:= \Sigma_V$, the Gel'fand spectrum of $V$ seen as a set, and for every inclusion $\iota_{W,V}:V \rightarrow W$ (morphism) "$x\stackrel{\iota_{W,V}}{\mapsto} x$" where $V,W$ in $\mathfrak{C}(\mathfrak{M})$ are such that $V \subseteq W$, we assign $\underline{\,\Sigma\,}(\iota_{V,W}):=res_{V,W}$ where $res_{V,W}:\Sigma_W \rightarrow \Sigma_V $ is the restriction in the Gel'fand spectrum: $$\begin{matrix} res_{V,W}:&\Sigma_W & \rightarrow & \Sigma_V\\ &\lambda & \mapsto & \lambda\hspace{-3pt}\upharpoonright_V \end{matrix}$$
Definition(Global Section): A global section $S$ for a presheaf $F$ ($F$ being a presheaf over a small category $C$) is also a presheaf ($S$ is a presheaf over $C$) such that for each $B$ in Obj$_C$, $S(B) =\{p_B\}$ for some point $p_B \in F(B)$ and $S(A \stackrel{f}{\rightarrow} B) \equiv \{p_B\}\stackrel{S(f)}{\rightarrow} \{p_A\}$ and where there are maps $\{ \iota_B\}_{B \text{ in } \text{Obj}_C}$ where each $\iota_B: S(B) \rightarrow F(B)$ with $\iota_B(p_B) = p_B$ is a collection that gives rise to a natural transformation.
In particular, a global section for the spectral presheaf $\underline{\,\Sigma\,}$ is a presheaf $X:\mathfrak{C}(\mathfrak{M}) \rightarrow \mathfrak{M}'$ that maps each object $V$ of $\mathfrak{C}(\mathfrak{M})$ (a context) into a linear functional $X(V) = \nu_V:V \rightarrow \mathbb{C}$ such that $\nu \in \underline{\,\Sigma\,}_V=\Sigma_V$, and that maps every morphism of $\mathfrak{C}(\mathfrak{M})$, i.e. the inclusion functions between contexts, $V \stackrel{\iota_{V,W}}{\rightarrow} W$ into $X(\iota_{V,W})=h:\{\nu_W\} \rightarrow \{\nu_V\}$, that is the function whose only domain point is the functional $\nu_W$ and whose only image is the functional $\nu_V$. Between any global section $X$ and the spectral presheaf $\underline{\,\Sigma\,}$, there is a collection of transformations $(i_V)_{V \in \mathfrak{C}(\mathfrak{M})}$ such that $i_V(X(V))=i_V(\nu_V)=\nu_V \in \Sigma_V=\underline{\,\Sigma\,}(V)$. These functions $i_V$ act like the components of a natural transformation (a category theory notion) between the presheaves (seen as functors) $X$ and $\underline{\,\Sigma\,}$. This naturality condition basically says that $i_V \circ X(\iota_{V,W}) = \underline{\,\Sigma\,}(\iota_{V,W}) \circ i_W$ for any $W \supset V;\,W,V \in \mathfrak{C}(\mathfrak{M})$. So that for every functional $\nu_W=X(W)$: $$i_V \circ X(\iota_{V,W})(\nu_W) = \underline{\,\Sigma\,}(\iota_{V,W}) \circ i_W (\nu_W)$$ $$i_V(\nu_V) = \underline{\,\Sigma\,}(\iota_{V,W}) (\nu_W)$$ $$\nu_V=\underline{\,\Sigma\,}(\iota_{V,W})(\nu_W)$$ $$\nu_V=res_{V,W}(\nu_W)$$ $$\nu_V=\nu_W\hspace{-3pt}\upharpoonright_V$$ This means that for every subset $U \subseteq V$, $X$ sends $U$ and $V$ to functionals that completely agree within the smallest subset. (hence the "global" adjective).
Then in the already cited article https://arxiv.org/pdf/quant-ph/9911020.pdf, a bijection between valuations and global sections of the Spectral presheaf is alluded to in page 6 (again changing the notation for a more contemporary one):
If it existed, a global element of $\underline{\,\Sigma\,}$ over $\mathfrak{C}(\mathfrak{M})$ would assign a multiplicative linear functional $\kappa : V → \mathbb{C}$ to each commutative von Neumann algebra V in $\mathfrak{C}(\mathfrak{M})$ in such a way that these functionals match up as they are mapped down the presheaf. To be precise, the functional $κ$ on $V$ would be obtained as the restriction to V of the functional $κ_1 : V_1 → \mathbb{C}$ for any $V_1 ⊇ V$. Furthermore, when restricted to the self-adjoint elements of $V$ , a multiplicative linear functional $κ$ satisfies all the conditions of a valuation, namely: (i) $\kappa(B) \in \sigma(B)$ and (ii) $\kappa(B) =\kappa(f(A)) = f(\kappa(A))$ whenever $B = f(A)$.
This is not actually proven in this insert or in the article. There was a minicurse in my university about this topic in which this bijection was proved for matrix algebras as a "simplifyied" version of the full bijection, namely:
Theorem: Let $v$ be a valuation on $(\mathbb{M}_n)_{sa}$ and $\varphi_\mathbf{C}^v$ be the unique character on the context $\mathbf{C}$ in $\mathfrak{C}((\mathbb{M}_n)_{sa})$ that extends $v$ (a lemma that I will call (1)). Then defining $X_v(\mathbf{C}) := \varphi_\mathbf{C}^v$ for each $\mathbf{C}$ in $\mathfrak{C}((\mathbb{M}_n)_{sa})$ defines a global section of $\underline{\,\Sigma\,}$. Furthermore, this assignment is bijective.
Proof: For a valuation $v$ on $(\mathbb{M}_n)_{sa}$, we first show that $X_v:\mathfrak{C}((\mathbb{M}_n)_{sa}) \rightarrow \underline{\,\Sigma\,}(\mathfrak{C}((\mathbb{M}_n)_{sa}))$ is a presheaf, firstly it does send contexts to characters, whose collection form a set for each valuation, and it will send inclusion morphisms through the contravariant spectral presheaf $\underline{\,\Sigma\,}$, which makes $X_v$ also contravariant and hence a presheaf. We shall now prove that $X_v(\mathbf{C})$ is indeed a global section, to do so we consider that for each morphism $\mathbf{C} \xrightarrow{f} \mathbf{D}$ the expression: $$\varphi_\mathbf{C}^v= \underline{\,\Sigma\,}(f)(\varphi_\mathbf{D}^v) = \varphi_\mathbf{D}^v \hspace{-3pt}\upharpoonright_\mathbf{C}$$ holds. It must hold since both $\varphi_\mathbf{C}^v$ and $\varphi_\mathbf{D}^v \hspace{-3pt}\upharpoonright_\mathbf{C}$ are characters on $\mathbf{C}$ that extend $v$ to $\mathbf{C}$, and such extensions are unique (1). We will now prove that this assignment is a bijection. Beginning by proving injectivity, we suppose $v \neq w$ are valuations. Since they are not equal, there is at least one element $A$ in $(\mathbb{M}_n)_{sa}$ such that $v(A) \neq w(A)$, hence $X_v(\mathcal{C}(A)) \neq X_w(\mathcal{C}(A))$, where $\mathcal{C}(A)$ is the context generated by $A\,\,\,\,\,$ ($\mathcal{C}(\{A\}) \equiv \mathcal{C}(A) $). For proving surjectivity we assume that $X$ is a a global section of $\underline{\,\Sigma\,}$. For each object $A$ in $(\mathbb{M}_n)_{sa}$ and considering characters $\varphi_\mathbf{C}$ obtained by $X(\mathbf{C}) = \{\varphi_\mathbf{C}\}$, we define $v_X(A):= \varphi_{\mathcal{C}(A)}(A)$. To show that $v_X$ is indeed a a valuation, we use a proposition (2) that tells us that $\mathcal{C}(A) =\{f(A) \, |\, f:\sigma(A) \rightarrow \mathbb{R} \}$, as a consequence $\mathcal{C}(f(A)) \subseteq \mathcal{C}(\{A\})$, meaning there is a unique morphism $f$ in $\textbf{Mor}_{\mathfrak{C}((\mathbb{M}_n)_{sa})}(\mathcal{C}(f(A)) , \mathcal{C}(A))$ (an inclusion morphism). As by hypothesis $X$ is a global section on $\underline{\,\Sigma\,}$, that means: $$\varphi_{\mathcal{C}(f(A))}=\underline{\,\Sigma\,}(f)(\varphi_{\mathcal{C}(A)}) =\varphi_{\mathcal{C}(A)}\hspace{-3pt}\upharpoonright_{\mathcal{C}(f(A))} $$ With these remarks in mind, notice that given an object $A$ in $(\mathbb{M}_n)_{sa}$ and a function $f:\mathbb{R} \rightarrow \mathbb{R}$, $$v_X(f(A)) = \varphi_{\mathcal{C}(f(A))}(f(A))=\varphi_{\mathcal{C}(A)} \hspace{-3pt}\upharpoonright_{\mathcal{C}(f(A))}(f(A))=\varphi_{\mathcal{C}(A)}(f(A)),$$ again by a proposition (3) that says that $\varphi(f(A)) = f(\varphi(A))$ for a function $f:\sigma(A) \rightarrow \mathbb{R}$, a character $\varphi \in \Sigma_{\mathcal{C}(A) }$ and $A \in (\mathbb{M}_n)_{sa}$, we get the following equation continuing from the last: $$\varphi_{\mathcal{C}(A)}(f(A)) = f(\varphi_{\mathcal{C}(A)}(A))= v_X(f(A)),$$ Hence $v_X(f(A))= v_X(f(A))$, making $v_X$ a valuation. If we now use $v_X$ to induce a global section, we’ll get back to $X$ due to the fact that lemma (1) ensures uniqueness of characters extending valuations. Indeed, notice that given an object $A$ in $(\mathbb{M}_n)_{sa}$, we have: $$\varphi_{\mathcal{C}(A)}^{v_X}(A) = v_X(A) = \varphi_{\mathcal{C}(A)}(A),$$ which ensures $\varphi_{\mathcal{C}(A)}^{v_X}$ and $\varphi_{\mathcal{C}(A)}$ agree on $\mathcal{C}(A)$, since characters commute with the functional calculus by (3) and by proposition (2) $\mathcal{C}(A) = \{f(A) \,|\, f:\sigma(A) \rightarrow \mathbb{R}\}\,$, however a proposition (4) tells us that all contexts $\mathbf{C}$ in $\mathfrak{C}((\mathbb{M}_n)_{sa})$ are of the form $\mathcal{C}(G) = \mathbf{C}$ for some $G \in (\mathbb{M}_n)_{sa}$, and hence the equality of $\varphi_\mathbf{C}^{v_X}$ and $\varphi_\mathbf{C}$ hold for all contexts $\mathbf{C} \in \mathfrak{C}((\mathbb{M}_n)_{sa})$.
I should note that in this proof 4 lemmas are used:
(1) There exists a unique character that extends a valuation $v$. (This is very straight forward to prove in either $\mathbb{M}_n$ or in more general von Neumann algebras, so this one is Ok.)
(2) $\mathcal{C}(A) = \{f(A) \,|\, f:\sigma(A) \rightarrow \mathbb{R}\}$; this I was able to prove for $\mathbb{M}_n$ using the property that in $\mathbb{M}_n$ spectral decompositions are formed of finite sums of projectors, then showing that each projector is in $\mathcal{C}(A)$ by: $$\mathcal{C}(\{A\}) \ni A-\alpha_2 = \sum\limits_{i=1}^n\alpha_iP_i-\alpha_2\sum\limits_{i=1}^n P_i =\sum\limits_{i=1}^n (\alpha_i-\alpha_2)P_i \in \mathcal{C}(\{A\})$$ $$\mathcal{C}(\{A\}) \ni \sum\limits_{i=1}^n (\alpha_i-\alpha_2)P_i-\alpha_3 = \sum\limits_{i=1}^n (\alpha_i-\alpha-\alpha_3)P_i \in \mathcal{C}(\{A\})$$ $$\vdots$$ $$\alpha_1P_1 \in \mathcal{C}(\{A\})$$ This makes any finite combination of projectors be in $\mathcal{C}(A)$ and hence $f(A) = \sum\limits_{i=1}^n f(\alpha_i) P_i$ so $\{f(A) \,|\, f:\sigma(A) \rightarrow \mathbb{R}\} \subseteq \mathcal{C}(\{A\})$, but $\{f(A) \,|\, f:\sigma(A) \rightarrow \mathbb{R}\}$ is itself trivially a context in $\mathbb{M}_n$ (that is for $A \in \mathbb{M}_n$), hence $\mathcal{C}(A) \subseteq \{f(A) \,|\, f:\sigma(A) \rightarrow \mathbb{R}\}$, making $\mathcal{C}(A) = \{f(A) \,|\, f:\sigma(A) \rightarrow \mathbb{R}\}$. In more general algebras I don't know how to prove this, although it was alluded in the minicurse that it was possible.
(3) For any $\varphi \in \Sigma_V$, $\varphi(f(A))= f(\varphi(A)),\, A \in V \subseteq \mathfrak{C}((\mathbb{M}_n)_{sa})$; this one I kinda got stuck proving even for $(\mathbb{M}_n)_{sa}$, the idea was to also expand $f(A)$ by the functional calculus and use the linearity of $\varphi$, but then I get to $\sum_{i=1}^n f(\alpha_i) \varphi(P_i)$, since $\varphi(P_i) \in \{0,1\} = \sigma(P_i)$ then this is just the sum of those $f(\alpha_i)$ for which $\varphi(P_i) \neq 0$, hence for linear $f$ one could get $f \left( \sum_{i=1}^n\alpha_i\varphi(P_i) \right) = f(\varphi(A))$, for more general algebras than $(\mathbb{M}_n)_{sa}$ I also don't know how to prove this, although it was again alluded to in the minicurse that this could be proven.
(4) $\forall \mathbf{C} \in \mathfrak{C}((\mathbb{M}_n)_{sa})$, $\,\,\mathbf{C}=\mathcal{C}(G)$ for some $G \in (\mathbb{M}_n)_{sa}$; This I was not able to prove at all for $(\mathbb{M}_n)_{sa}$ and of all the propositions this was the only one for which the professor explicitly said it was not possible to prove for more general algebras even for some algebras that appear in $QM$ (I don't know if he was meaning that it isn't valid for $C^*$-algebras or for von Neumann algebras or whatnot, he also said thatthis one was way more technical to prove even in $(\mathbb{M}_n)_{sa}$ and so he did not give any idea of the proof.).
My idea was to use the same argument as the one used on the proof of the Theorem above for more general von Neumann algebras instead of the "available" proof for $(\mathbb{M}_n)_{sa}$, trying to exchange these four lemmas for versions valid on von Neumann algebras and possibly lifting the necessity of $f$ being continuous for it being a limited Borel measurable function since for those the functional calculus $f(A)$ is well defined for self-adjoint $A$ (even for normal $A$). Such a generalization must be possible since in the original paper this bijection between global sections and valuations was basically asserted as was already shown in the quote of page 6 of https://arxiv.org/pdf/quant-ph/9911020.pdf.
Sorry for the extension of the question, but I feel that I have to clearly specify everything otherwise people begin to point out that something seems badly defined, I would very much appreciate if anyone could help me with this.
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UPDATE: After a answer from @Plop in a related question, I have come to the conclusion that my working definition of a context is imprecise. Also, by re-reading the original papers, in all physically relevant situations they use that the context of a von Neumann algebra is a sub-von Neumann algebra. Hence, the definition should be changed to: "For a algebra $\mathcal{A}$ that is at least a unital $*$-algebra, a context of $\mathcal{A}$ is a sub-algebra of $\mathcal{A}$ that is a algebra of the same kind of $\mathcal{A}$, and such that the units coincide.".
For all my proposes I then decided that proving (2)-(4) for concrete von Neumann algebras is sufficient. Then using the discussion with @Plop I completed a proof of the (2) assertion, it goes like this:
(2) -proof-> For a concrete von Neumann algebra $\mathfrak{A}$, the generated context of a self-adjoint operator $A\in \mathfrak{A}$ is the von Neumann algebra $\mathcal{C}(A)$, since the set $\{ \, f(A) \, \in \mathfrak{A} \, | \, f \text{ a bounded Borel measurable function } \}$ commutes with $A$ by the Borel functional calculus for bounded Borel measurable functions, then that set is a subset of $\mathcal{C}(A)'$ that is a subset of $\mathcal{C}(A)''= \mathcal{C}(A)$ by the bicomutant theorem, so we get one containment.
To get the reciprocal we consider that by the Borel functional calculus for bounded Borel measurable functions (or at least by the enunciation of this theorem that I am using, which includes the following assertion) the set $\{ \, f(A) \, \in \mathfrak{A} \, | \, f\text{ a bounded Borel measurable function } \}$ is strong operator closed, hence since for bounded operators on a Hilbert space I can show that strong operator closure implies weak operator closure, using the polarisation identity, then the set with the functional calculi is a abelian von Neumann algebra, again by the Borel functional calculi, and hence a context contained in the generated context of $A$ by the previous paragraph, then $\{ \, f(A) \, \in \mathfrak{A} \, | \, f \text{ a bounded Borel measurable function } \} = \mathcal{C}(A)$; otherwise there would be a smaller context containing $A$ than the one generated by $A$.
It still remais to prove assertions $(3)$ and $(4)$ for concrete von Neumann algebras. For proving $(3)$ I am convinced that I just need to use either a polinomial approximation by Stone-Weierstrass and then use Lusin's theorem to generalise this for bounded Borel measurable functions, or to use that every continuous function can be approximated by a sequence of simple functions, and then again use Lusin's theorem, for then to use that those characters from the Gel'fand spectrum are continuous to get the commutativity of characters with bounded Borel measurable functions, the only problem is that although I seem to get or a polinomial or a simple function by applying this idea, at the end it is not clear why those are the flipped order composition.
For assertion (3) I still have no Idea, even for a matrix algebra.