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\begin{document}

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\section*{Harding-Döring-Hamhalter theorem}

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

\hypertarget{operator_algebra_and_aqft}{}\paragraph*{{Operator algebra and AQFT}}\label{operator_algebra_and_aqft}

[[!include AQFT and operator algebra contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statements}{Statements}\dotfill \pageref*{statements} \linebreak
\noindent\hyperlink{related_theorems}{Related theorems}\dotfill \pageref*{related_theorems} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Theorems observed by Harding, D\"o{}ring and Hamhalter say that under some conditions the [[Jordan algebra]] structure of a [[C\emph{-algebra]] is effectively captured by its [[poset of commutative subalgebras]].}

Via the [[Alfsen-Shultz theorem]], which states that two [[C\emph{-algebra]] have the same [[states]] precisely if they are isomorphic as [[Jordan algebras]], this is related to [[Gleason's theorem]], which says that states on an [[algebra of bounded operators]] are determined by their restriction to commutative subalgebras (if the underlying Hilbert space has dimenion $\gt 2$).}

\hypertarget{statements}{}\subsection*{{Statements}}\label{statements}

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 (1975), 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 10}).

There is a generalization of this theorem to more general [[C-star algebras]] in (\hyperlink{Hamhalter}{Hamhalter 11}).

\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{related_theorems}{}\subsection*{{Related theorems}}\label{related_theorems}

Other theorems about the foundations and [[interpretation of quantum mechanics]] include:

\begin{itemize}%
\item [[order-theoretic structure in quantum mechanics]]

\begin{itemize}%
\item [[Kochen-Specker theorem]]


\item [[Alfsen-Shultz theorem]]



\end{itemize}

\item [[Fell's theorem]]


\item [[Gleason's theorem]]


\item [[Wigner theorem]]


\item [[Bell's theorem]]


\item [[Bub-Clifton theorem]]


\item [[Kadison-Singer problem]]



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

The 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 (\href{http://dx.doi.org/10.1016/j.jmaa.2011.05.035}{journal})

\end{itemize}
\begin{itemize}%
\item [[Jan Hamhalter]], E. Turilova, \emph{Structure of associative subalgebras of Jordan operator algebras} (\href{http://arxiv.org/abs/1111.7240}{arXiv:1111.7240})

\end{itemize}


\end{document}