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\begin{document}

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\section*{poset of commutative subalgebras}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{$(0,1)$-Category theory}}\label{category_theory}

[[!include (0,1)-category theory - contents]]

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{operator_algebra}{}\paragraph*{{Operator algebra}}\label{operator_algebra}

[[!include AQFT and operator algebra contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{as_a_site_for_noncommutative_geometry}{As a site for noncommutative geometry}\dotfill \pageref*{as_a_site_for_noncommutative_geometry} \linebreak
\noindent\hyperlink{as_a_site_for_noncommutative_phase_spaces}{As a site for noncommutative phase spaces}\dotfill \pageref*{as_a_site_for_noncommutative_phase_spaces} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak
\noindent\hyperlink{RelationToJordanAlgebras}{Relation to Jordan algebras}\dotfill \pageref*{RelationToJordanAlgebras} \linebreak
\noindent\hyperlink{the_presheaf_topos_over_}{The presheaf topos over $ComSub(A)^{op}$}\dotfill \pageref*{the_presheaf_topos_over_} \linebreak
\noindent\hyperlink{the_locale_}{The locale $\Sigma(A)$}\dotfill \pageref*{the_locale_} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general_3}{General}\dotfill \pageref*{general_3} \linebreak
\noindent\hyperlink{ReferencesRelationToJordanAlgebras}{Relation to Jordan algebras}\dotfill \pageref*{ReferencesRelationToJordanAlgebras} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

For $A$ a an [[associative algebra]], not necessarily [[commutative algebra|commutative]], its collection $ComSub(A)$ of commutative subalgebras $B \hookrightarrow A$ is naturally a [[poset]] under inclusion of subalgebras.

\hypertarget{as_a_site_for_noncommutative_geometry}{}\subsubsection*{{As a site for noncommutative geometry}}\label{as_a_site_for_noncommutative_geometry}

Various authors have proposed (\hyperlink{ButterfieldHamiltonIsham}{Butterfield-Hamilton-Isham}, \hyperlink{DoeringIsham}{D\"o{}ring-Isham}, \hyperlink{HeunenLandsmanSpitters}{Heunen-Landsmann-Spitters}) that for the case that $A$ is a [[C-star algebra]] the [[noncommutative geometry]] of the [[Isbell duality|formal dual]] [[space]] $\Sigma(A)$ of $A$ may be understood as a commutative [[geometry]] [[internalization|internal]] to a [[sheaf topos]] $\mathcal{T}_A$ over $ComSub(A)$ or its [[opposite category|opposite]] $ComSub(A)^{op}$. An advantage of the latter is that $\Sigma$ becomes a [[compact locale|compact]] [[regular locale]].

\hypertarget{as_a_site_for_noncommutative_phase_spaces}{}\subsubsection*{{As a site for noncommutative phase spaces}}\label{as_a_site_for_noncommutative_phase_spaces}

Specifically, consider the case that the algebra $A = B(\mathcal{H})$ is that of [[bounded operator]]s on a [[Hilbert space]]. This is interpreted as an [[algebra of quantum observables]] and the commutative subalgebras are then ``[[classical contexts]]''.

Applying [[Bohrification]] to this situation (see there for more discussion), one finds that the [[locale]] $\Sigma(A)$ internal to $\mathcal{T}_A$ behaves like the noncommutative [[phase space]] of a system of [[quantum mechanics]], which however internally looks like an ordinary commutative geometry. Various statements about [[operator algebra]] then have geometric analogs in $\mathcal{T}_A$.

Notably the [[Kochen-Specker theorem]] says that $\Sigma(B(\mathcal{H}))$, while nontrivial, has no [[point]]s/no [[global element]]s. (This topos-theoretic geometric reformulation of the Kochen-Specker theorem had been the original motivation for considering $ComSub(A)$ in the first place in \hyperlink{ButterfieldIsham}{ButterfieldIsham}).

Moreover, inside $\mathcal{T}_A$ the [[quantum mechanics|quantum mechanical]] kinematics encoded by $B(\mathcal{H})$ looks like [[classical mechanics]] kinematics internal to $\mathcal{T}_A$ (\hyperlink{HeunenLandsmanSpitters}{HeunenLandsmannSpitters}, following \hyperlink{DoeringIsham}{D\"o{}ringIsham}):

\begin{enumerate}%
\item the [[open subset]]s of $\Sigma(A)$ are identified with the quantum [[state]]s on $A$. Their collection forms the [[Heyting algebra]] of quantum logic.


\item [[observable]]s are morphisms of internal [[locale]]s $\Sigma(A) \to IR$, where $IR$ is the [[interval domain]].



\end{enumerate}
The assignment to a noncommutative algebra $A$ of a [[locale]] $\underline{\Sigma}_A$ internal to $\mathcal{T}_A$ has been called [[Bohrification]], in honor of [[Nils Bohr]] whose heuristic writings about the nature of [[quantum mechanics]] as being probed by classical (= commutative) context one may argue is being formalized by this construction.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{general_2}{}\subsubsection*{{General}}\label{general_2}

\begin{lemma}
\label{}\hypertarget{}{}
The poset of commutative subalgebras $C(A)$ is always an (unbounded) meet-[[semilattice]]. If $A$ itself is commutative then it is a bounded meet semilattice, with $A$ itself being the top element.

\end{lemma}
\hypertarget{RelationToJordanAlgebras}{}\subsubsection*{{Relation to Jordan algebras}}\label{RelationToJordanAlgebras}

For $A$ an [[associative algebra]] write $A_J$ for its corresponding [[Jordan algebra]], where the commutative product $\circ : A_J \otimes A_J \to A_J$ is the symmetrization of the product in $A$: $a \circ b = \frac{1}{2}(a b + b a)$.

\begin{lemma}
\label{}\hypertarget{}{}
There exist von Neumann algebras $A$, $B$ such that there exists a Jordan algebra isomorphism $A_J \to B_J$ but not an algebra isomorphism $A \to B$.

\end{lemma}
\begin{proof}
By

\begin{itemize}%
\item [[Alain Connes]], \emph{A factor not anti-isomorphic to itself}, Annals of Mathematics, 101 (1962), no. 3, 536--554. (\href{http://www.jstor.org/stable/1970940}{JSTOR})

\end{itemize}
there is a [[von Neumann algebra factor]] $A$ with no algebra isomorphism to its opposite algebra $A^{op}$. But clearly $A_J \simeq (A^{op})_J$.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Let $A, B$ be [[von Neumann algebra]]s without a type $I_2$-[[von Neumann algebra factor]]-summand and let $ComSub(A)$, $ComSub(B)$ be their posets of [[commutative C-star-algebra|commutative]] sub-von Neumann algebras.

Then every [[isomorphism]] $ComSub(A) \to ComSub(B)$ of [[poset]]s comes from a unique [[Jordan algebra]] isomorphism $A_J \to B_J$.

\end{prop}
This is the theorem in (\hyperlink{HardingDoering}{Harding-D\"o{}ring}).

There is a generalization of this theorem to more general [[C-star algebras]] in (\hyperlink{Hamhalter}{Hamhalter}).

For more on this see at \emph{[[Harding-Döring-Hamhalter theorem]]}.

\begin{remark}
\label{}\hypertarget{}{}
This is related to the [[Alfsen-Shultz theorem]], which says that two $C^*$-algebras have the same [[state on an operator algebra|states]] precisely if they are Jordan-isomorphic.

\end{remark}
\hypertarget{the_presheaf_topos_over_}{}\subsubsection*{{The presheaf topos over $ComSub(A)^{op}$}}\label{the_presheaf_topos_over_}

\begin{defn}
\label{}\hypertarget{}{}
For $A$ a [[C-star algebra]], write $ComSub(A)$ for its [[poset]] of sub-$C^*$-algebras. Write

\begin{displaymath}
\mathcal{T}_A := [ComSub(A),Set]
\end{displaymath}
for the [[presheaf topos]] on $ComSub(A)^{op}$. This is alse called the \emph{[[Bohr topos]]}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This opposite order on commutative subalgebras may be seen as the information order from [[Kripke semantics]]: a larger subalgebra contains more information. In this light the presheaf topos on $ComSub(A)$, as used by (\hyperlink{DoeringIsham}{D\"o{}ring-Isham 07}) and co-workers, may be seen as the co-Kripke model. This model is also referred to as the \emph{coarse-graining semantics} of quantum mechanics. See also at \emph{[[spectral presheaf]]}.

\end{remark}
\begin{lemma}
\label{}\hypertarget{}{}
The topos $\mathcal{T}_A$ is a [[localic topos]].

\end{lemma}
Because $ComSub(A)$ is a [[posite]].

\hypertarget{the_locale_}{}\subsubsection*{{The locale $\Sigma(A)$}}\label{the_locale_}

\begin{prop}
\label{JordanIsosAndCSubIsos}\hypertarget{JordanIsosAndCSubIsos}{}
The [[presheaf]]

\begin{displaymath}
(\mathbb{A} : B \mapsto U(B)) \;\; \in \mathcal{T}_A
  \,,
\end{displaymath}
where $U(B)$ is the underlying [[set]] of the commutative subalgebra $B$, is canonically a commutative $C^*$-algebra [[internalization|internal]] to $\mathcal{T}_A$.

\end{prop}
This is (\hyperlink{HeunenLandsmanSpitters}{HeunenLandsmanSpitters, theorem 5}).

\begin{cor}
\label{}\hypertarget{}{}
By the [[constructive Gelfand duality theorem]] there is uniquely a [[locale]] $\Sigma(A)$ internal to $\mathcal{T}_A$ such that $\mathbb{A}$ is the internal commutative $C^*$-algebra of functions on $\Sigma(A)$.

\end{cor}
This observation is amplified in (\hyperlink{HeunenLandsmanSpitters}{HeunenLandsmanSpitters}).

\begin{prop}
\label{}\hypertarget{}{}
If $A = \mathcal{B}(H)$ is the algebra of [[bounded operator]]s on a [[Hilbert space]] $H$ of [[dimension]] $\gt 2$, then then [[Kochen-Specker theorem]] implies that $\Sigma(A)$ has no points/no [[global element]].

\end{prop}
This is (\hyperlink{HeunenLandsmanSpitters}{HeunenLandsmanSpitters, theorem 6}), following (\hyperlink{ButterfieldHamiltonIsham}{Butterfield-Hamilton-Isham}).

\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general_3}{}\subsubsection*{{General}}\label{general_3}

The proposal that the the noncommutative geometry of $A$ is fruitfully studied via the commutative geometry over $ComSub(A)$ goes back to

\begin{itemize}%
\item [[Jeremy Butterfield]], John Hamilton, [[Chris Isham]], \emph{A topos perspective on the Kochen-Specker theorem}

\emph{I. quantum states as generalized valuations} International Journal of Theoretical Physics, 37(11):2669--2733, 1998.

\emph{II. conceptual aspects and classical analogues} International Journal of Theoretical Physics, 38(3):827--859, 1999

\emph{III. Von Neumann algebras as the base category} International Journal of Theoretical Physics, 39(6):1413--1436, 2000.



\end{itemize}
The proposal that the non-commutativity of the [[phase space]] in [[quantum mechanics]] is fruitfully understood in the light of this has been amplified in a series of articles

\begin{itemize}%
\item [[Andreas Döring]], [[Chris Isham]], \emph{A topos foundation for theories of physics} Journal of Mathematical Physics (2008)

\end{itemize}
The presheaf topos on $ComSub(A)^{op}$ ([[Bohr topos]]) and its internal localic Gelfand dual to $A$ is discussed in

\begin{itemize}%
\item [[Chris Heunen]], [[Klaas Landsman]], [[Bas Spitters]], \emph{A topos for algebraic quantum theory} (\href{http://arxiv.org/abs/0709.4364}{arXiv:0709.4364})

\end{itemize}
See also [[higher category theory and physics]].

\hypertarget{ReferencesRelationToJordanAlgebras}{}\subsubsection*{{Relation to Jordan algebras}}\label{ReferencesRelationToJordanAlgebras}

The \hyperlink{RelationToJordanAlgebras}{relation to Jordan algebras} of $ComSub(A)$ is discussed in

\begin{itemize}%
\item [[John Harding]], [[Andreas Döring]], \emph{Abelian subalgebras and the Jordan structure of a von Neumann algebra} (\href{http://arxiv.org/abs/1009.4945}{arXiv:1009.4945})

\end{itemize}
for $A$ a [[von Neumann algebra]] and more generally for $A$ a [[C\emph{-algebra]] in}

\begin{itemize}%
\item [[Jan Hamhalter]], \emph{Isomorphisms of ordered structures of abelian $C^\ast$-subalgebras of $C^\ast$-algebras}, J. Math. Anal. Appl. 383 (2011) 391--399, doi:\href{https://dx.doi.org/10.1016/j.jmaa.2011.05.035}{10.1016/j.jmaa.2011.05.035}

\end{itemize}
See at \emph{[[Harding-Döring-Hamhalter theorem]]}.

[[!redirects semilattice of commutative subalgebras]]



\end{document}