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\begin{document}

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\section*{Gelfand-Naimark-Segal construction}

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

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{for_algebras}{For $C^\ast$-algebras}\dotfill \pageref*{for_algebras} \linebreak
\noindent\hyperlink{ForCStarCategories}{For $C^\ast$-categories}\dotfill \pageref*{ForCStarCategories} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{Gelfand-Naimark-Segal construction} (``GNS construction'') represents a [[state on a star-algebra]] over the [[complex numbers]], which a priori is defined purely algebraically as a non-degenerate positive linear function

\begin{displaymath}
\rho \;\colon\; \mathcal{A} \longrightarrow \mathbb{C}
  \,,
\end{displaymath}
by a [[vector]] $\psi \in \mathcal{H}$ in a [[complex numbers|complex]] [[Hilbert space]] $\mathcal{H}$ as the ``[[expectation value]]''

\begin{displaymath}
\begin{aligned}
    \rho(A) 
    & = \langle \psi \vert \, A \, \vert \psi \rangle
    \\
    & \coloneqq \langle \psi, \pi(A) \psi \rangle  
  \end{aligned}
\end{displaymath}
with respect to some [[star-representation]]

\begin{displaymath}
\pi \;\colon\; \mathcal{A} \longrightarrow End(\mathcal{H})
\end{displaymath}
of $\mathcal{A}$ on (a [[dense subspace]] of) $\mathcal{H}$; where $\langle -,-\rangle$ denotes the [[inner product]] on the [[Hilbert space]].

Originally this was considered for [[C\emph{-algebras]] and [[C}-representations]] (\hyperlink{GelfandNaimark43}{Gelfand-Naimark 43}, \hyperlink{Segal47}{Segal 47}), see for instance (\hyperlink{Schmuedgen90}{Schm\"u{}dgen 90}), but the construction applies to general [[unital algebra|unital]] [[star algebras]] $\mathcal{A}$ (\hyperlink{KhavkineMoretti15}{Khavkine-Moretti 15}) as well as to other coefficient rings, such as to [[formal power series algebras]] over $\mathbb{C}[ [\hbar] ]$ (\hyperlink{BordemannWaldmann96}{Bordemann-Waldmann 96}).

The GNS-construction plays a central role in [[algebraic quantum field theory]] (\hyperlink{Haag96}{Haag 96}, \hyperlink{Moretti18}{Moretti 18}, \hyperlink{KhavkineMoretti15}{Khavkine-Moretti 15}), where $\mathcal{A}$ plays the role of an [[algebra of observables]] and $\rho \colon \mathcal{A} \to \mathbb{C}$ the role of an actual state of a [[physical system]] (whence the terminology) jointly constituting the ``[[Heisenberg picture]]''-perspective of [[quantum physics]]; so that the GNS-construction serves to re-construct a corresponding [[Hilbert space|Hilbert]] [[space of states]] as in the [[Schrödinger picture]] of quantum physics. In this context the version for [[C\emph{-algebras]] corresponds to [[non-perturbative quantum field theory]], while the generalization to [[formal power series algebras]] corresponds to [[perturbative quantum field theory]].}

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\begin{quote}%
under construction


\end{quote}
\hypertarget{for_algebras}{}\subsubsection*{{For $C^\ast$-algebras}}\label{for_algebras}

\begin{theorem}
\label{}\hypertarget{}{}
Given

\begin{enumerate}%
\item a [[C\emph{-algebra]], $\mathcal{A}$;}


\item a [[states in AQFT and operator algebra|state]], $\rho \;\colon\; \mathcal{A} \to \mathbb{C}$



\end{enumerate}
there exists

\begin{enumerate}%
\item a [[C\emph{-representation]]}

\begin{displaymath}
\pi 
  \;\colon\;
  \mathcal{A}
   \longrightarrow
   End(\mathcal{H})
\end{displaymath}
of $\mathcal{A}$ on some [[Hilbert space]] $\mathcal{H}$


\item a [[cyclic vector]] $\psi \in \mathcal{H}$



\end{enumerate}
such that $\rho$ is the state corresponding to $\psi$, in that

\begin{displaymath}
\begin{aligned}
    \rho(A) 
      & = \langle \psi \vert\, A \, \vert \psi \rangle
      \\
      & \coloneqq
      \langle \psi , \pi(A) \psi \rangle
  \end{aligned}
\end{displaymath}
for all $A \in \mathcal{A}$.

\end{theorem}
\hypertarget{ForCStarCategories}{}\subsubsection*{{For $C^\ast$-categories}}\label{ForCStarCategories}

The GNS construction for $C^\ast$-algebras is a special case of a more general construction of Ghez, Lima and Roberts applied to [[C\emph{-categories]] ([[horizontal categorification]] of $C^\ast$-algebras).}

\begin{theorem}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a $C^\ast$-category. Fix an object $A \in \operatorname{Ob}\mathcal{C}$ and let $\sigma$ be a state on the $C^\ast$-algebra $\mathcal{C}(A,A)$. Then there exists a $*$-representation

\begin{displaymath}
\rho_\sigma \colon \mathcal{C} \to \mathbf{Hilb}
\end{displaymath}
together with a cyclic vector $\xi \in \rho_\sigma(A)$ such that for all $x \in \mathcal{C}(A,A)$,

\begin{displaymath}
\sigma(x) = \langle \xi, \rho_\sigma(x)\xi \rangle.
\end{displaymath}
\end{theorem}
A [[C\emph{-algebra]] $\mathcal{A}$ is a $C^\ast$-category with a single [[object]] $\bullet$, where we make the identification $A = \mathcal{A}(\bullet,\bullet)$. In this case the theorem reduces to the classical GNS construction.}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Riesz representation theorem]]

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

The original construction for [[C\emph{-algebras]] and [[C}-representations]] is due to

\begin{itemize}%
\item [[Israel Gelfand]], [[Mark Naimark]], \emph{On the imbedding of normed rings into the ring of operators on a Hilbert space}. Matematicheskii Sbornik. 12 (2): 197--217 (1943)


\item [[Irving Segal]], \emph{Irreducible representations of operator algebras} (\href{http://www.ams.org/journals/bull/1947-53-02/S0002-9904-1947-08742-5/S0002-9904-1947-08742-5.pdf}{pdf}). Bull. Am. Math. Soc. 53: 73--88, (1947)



\end{itemize}
see for instance

\begin{itemize}%
\item K. Schm\"u{}dgen, \emph{Unbounded operator algebras and representation theory}, Operator theory, advances and applications, vol. 37. Birkh\"a{}user, Basel (1990)

\end{itemize}
The application to [[algebraic quantum field theory]] is discussed in

\begin{itemize}%
\item [[Rudolf Haag]], \emph{Local Quantum Physics: Fields, Particles, Algebras}, Texts and Monographs in Physics. Springer (1996).


\item [[Valter Moretti]], \emph{Spectral Theory and Quantum Mechanics :Mathematical Structure of Quantum Theories, Symmetries and introduction to the Algebraic Formulation}, 2nd ed. Springer Verlag, Berlin (2018)



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark%E2%80%93Segal_construction}{Gelfand-Naimark-Segal construction}}

\end{itemize}
For general unital [[star-algebras]]:

\begin{itemize}%
\item [[Igor Khavkine]], [[Valter Moretti]], \emph{Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction}, Chapter 5 in [[Romeo Brunetti]] et al. (eds.) \emph{Advances in Algebraic Quantum Field Theory}, , Springer, 2015

\end{itemize}
in relation with the classical moment problem and the notion of [[POVM]]

\begin{itemize}%
\item Nicolò Drago, [[Valter Moretti]], The notion of observable and the moment problem for \emph{-algebras and their GNS representations (\href{https://arxiv.org/abs/1903.07496}{arXiv.org:1903.07496})}

\end{itemize}
For [[formal power series algebras]] over $\mathbb{C}[ [ \hbar ] ]$:

\begin{itemize}%
\item [[Martin Bordemann]], [[Stefan Waldmann]], \emph{Formal GNS Construction and States in Deformation Quantization}, Commun. Math. Phys. (1998) 195: 549. (\href{https://arxiv.org/abs/q-alg/9607019}{arXiv:q-alg/9607019}, \href{https://doi.org/10.1007/s002200050402}{doi:10.1007/s002200050402})

\end{itemize}
Discussion in terms of [[universal properties]] in ([[higher category theory|higher]]) [[category theory]] is in

\begin{itemize}%
\item [[Bart Jacobs]], \emph{Involutive Categories and Monoids, with a GNS-correspondence}, Foundations of Physics, July 2012, Volume 42, Issue 7, pp 874--895 (\href{https://arxiv.org/abs/1003.4552}{arXiv:1003.4552})


\item [[Arthur Parzygnat]], \emph{From observables and states to Hilbert space and back: a 2-categorical adjunction} (\href{https://arxiv.org/abs/1609.08975}{arXiv:1609.08975})



\end{itemize}
category: operator algebras

[[!redirects Gelfand-Naimark-Segal construction]] [[!redirects Gelfand–Naimark–Segal construction]] [[!redirects Gelfand--Naimark--Segal construction]] [[!redirects GNS construction]] [[!redirects GNS-construction]] [[!redirects GNS representation]] [[!redirects GNS-representation]] [[!redirects GNS theorem]]



\end{document}