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\begin{document}

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\section*{state on a star-algebra}

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

\hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory}

[[!include measure theory - contents]]

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

[[!include functional analysis - contents]]

\hypertarget{algeraic_quantum_field_theory}{}\paragraph*{{Algeraic Quantum Field Theory}}\label{algeraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

\begin{quote}%
This entry describes a concrete formalization of the general notion of [[state]] in the context of [[AQFT]] and [[operator algebra]].


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

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{for_unital_star_algebras}{For unital star algebras}\dotfill \pageref*{for_unital_star_algebras} \linebreak
\noindent\hyperlink{for_algebras}{For $C^\ast$-Algebras}\dotfill \pageref*{for_algebras} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{exposition}{Exposition}\dotfill \pageref*{exposition} \linebreak
\noindent\hyperlink{original_articles}{Original articles}\dotfill \pageref*{original_articles} \linebreak


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

In [[physics]], a \emph{[[state]]} $\langle - \rangle$ is the information that allows to assign to each [[observable]] $A$ the [[expectation value]] $\langle A\rangle$ that this observable has when the [[physical system]] is assumed to be in that state.

Often this is formalized in the [[Schrödinger picture]] where a [[Hilbert space|Hilbert]] [[space of states]] $\mathcal{H}$ is taken as primary, and the [[observables]] are [[representation|represented]] as suitable [[linear operators]] $A$ on $\mathcal{H}$. Then for $\psi \in \mathcal{H}$ a state ([[pure state]]) the [[expectation value]] of $A$ in this state is the [[inner product]] $\langle \psi \vert A \vert \psi \rangle \coloneqq (\psi, A \psi)$. This defines a [[linear function]]

\begin{displaymath}
\langle \psi \vert (-) \vert \psi \rangle
  \;\colon\;
  \mathcal{A}
  \longrightarrow
  \mathbb{C}
\end{displaymath}
on the [[algebra of observables]] $\mathcal{A}$, satisfying some extra properties.

Conversely, in the [[Heisenberg picture]] one may take the ``abstract'' [[associative algebra|algebra]] [[algebra of quantum observables|of observables]] as primary (i.e. not necessarily manifested as an [[operator algebra]]), and declare that a state is any linear functional

\begin{displaymath}
\langle - \rangle
  \;\colon\;
  \mathcal{A}
  \longrightarrow
  \mathbb{C}
\end{displaymath}
which is \emph{positive} in that $\langle A^\ast A\rangle \geq 0$ and \emph{normalized} in that $\langle 1\rangle = 1$. Under suitable conditions a [[Hilbert space|Hilbert]] [[space of states]] may be (re-)constructed from this \emph{a posteriori} via the \emph{[[GNS construction]]}.

Traditionally this definition is considered for [[algebras of observables]] which are [[C\emph{-algebras]] (as usually required for [[non-perturbative quantum field theory]], see e.g. \hyperlink{Fredenhagen03}{Fredenhagen 03, section 2}), but the definition makes sense generally for plain [[star-algebras]], such as for instance for the [[formal power series algebras]] that appear in [[perturbative quantum field theory]] (e.g. \hyperlink{BordemannWaldmann96}{Bordemann-Waldmann 96, def. 1}, \hyperlink{FredenhagenRejzner12}{Fredenhagen-Rejzner 12, def. 2.4}, \hyperlink{KhavkineMoretti15}{Khavkine-Moretti 15, def. 6}, \hyperlink{Duetsch18}{Dütsch 18, def. 2.11}).}

The perspective that states are normalized positive linear functionals on the algebra of observables is implicit in traditional [[perturbative quantum field theory]], where it is encoded in the [[2-point function]] corresponding to a [[vacuum state]] or more generally a [[quasi-free quantum state]] (the \emph{[[Hadamard propagator]]}). The perspective is made explicit in [[algebraic quantum field theory]] (see e.g. \hyperlink{Fredenhagen03}{Fredenhagen 03, section 2}) and for [[star-algebras]] of observables that are not necessarily [[C\emph{-algebras]] in [[perturbative algebraic quantum field theory]] (e.g. \hyperlink{BordemannWaldmann96}{Bordemann-Waldmann 96}, \hyperlink{FredenhagenRejzner12}{Fredenhagen-Rejzner 12, def. 2.4}, \hyperlink{KhavkineMoretti15}{Khavkine-Moretti 15, def. 6}, \hyperlink{Duetsch18}{Dütsch 18, def. 2.11}).}

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{for_unital_star_algebras}{}\subsubsection*{{For unital star algebras}}\label{for_unital_star_algebras}

\begin{defn}
\label{StateOnAStarAlgebra}\hypertarget{StateOnAStarAlgebra}{}
\textbf{([[state]] on a [[unital algebra|unital]] [[star algebra]])}

Let $\mathcal{A}$ be a [[unital]] [[star-algebra]] over the [[complex numbers]] $\mathbb{C}$. A \emph{state} on $\mathcal{A}$ is a [[linear function]]

\begin{displaymath}
\rho \;\colon\; \mathcal{A} \longrightarrow \mathbb{C}
\end{displaymath}
such that

\begin{enumerate}%
\item (positivity) for all $A \in \mathcal{A}$ the value of $\rho$ on the product $A^\ast A$ is

\begin{enumerate}%
\item [[real part|real]] $\rho(A^\ast A) \in \mathbb{R} \hookrightarrow \mathbb{C}$


\item as such [[non-negative real number|non-negative]]: $\rho(A^\ast A) \geq 0$.



\end{enumerate}

\item (normalization) $\rho(1) = 1$.



\end{enumerate}
\end{defn}
(e.g. \hyperlink{BordemannWaldmann96}{Bordemann-Waldmann 96}, \hyperlink{FredenhagenRejzner12}{Fredenhagen-Rejzner 12, def. 2.4}, \hyperlink{KhavkineMoretti15}{Khavkine-Moretti 15, def. 6})

\begin{remark}
\label{}\hypertarget{}{}
\textbf{([[probability theory|probability theoretic]] interpretation of [[state on a star-algebra]])}

A [[star algebra]] $\mathcal{A}$ equipped with a [[state on a star-algebra|state]] is also called a \emph{[[quantum probability space]]}, at least when $\mathcal{A}$ is in fact a [[von Neumann algebra]].

\end{remark}
\begin{remark}
\label{StatesFormAConvexSet}\hypertarget{StatesFormAConvexSet}{}
\textbf{([[state on a star-algebra|states]] form a [[convex set]])}

For $\mathcal{A}$ a unital [[star-algebra]], the [[set]] of states on $\mathcal{A}$ according to def. \ref{StateOnAStarAlgebra} is naturally a [[convex set]]: For $\rho_1, \rho_2 \colon \mathcal{A} \to \mathbb{C}$ two states then for every $p \in [0,1] \subset \mathbb{R}$ also the [[linear combination]]

\begin{displaymath}
\itexarray{
    \mathcal{A}
     &\overset{p \rho_1 + (1-p) \rho_2}{\longrightarrow}&
    \mathbb{C}
    \\
    A &\mapsto& p \rho_1(A) + (1-p) \rho_2(A)
  }
\end{displaymath}
is a state.

\end{remark}
\begin{defn}
\label{PureStateOnAStarAlgebra}\hypertarget{PureStateOnAStarAlgebra}{}
\textbf{([[pure state]])}

A state $\rho \colon \mathcal{A} \to \mathbb{C}$ on a unital [[star-algebra]] (def. \ref{StateOnAStarAlgebra}) is called a \emph{[[pure state]]} if it is extremal in the [[convex set]] of all states (remark \ref{StatesFormAConvexSet}) in that an identification

\begin{displaymath}
\rho = p \rho_1 + (1-p) \rho_2
\end{displaymath}
for $p \in (0,1)$ implies that $\rho_1 = \rho_2$ (hence $= \rho$).

\end{defn}
\hypertarget{for_algebras}{}\subsubsection*{{For $C^\ast$-Algebras}}\label{for_algebras}

The following discusses states specifically on [[C\emph{-algebras]].}

\begin{defn}
\label{}\hypertarget{}{}
An element $A$ of an (abstract) $C^*$-algebra is called \textbf{[[positive operator|positive]]} if it is [[self-adjoint operator|self-adjoint]] and its [[spectrum of an operator|spectrum]] is contained in $[0, \infinity)$. We write $A \ge 0$ and say that the set of all [[positive operators]] is the positive cone (of a given $C^*$-algebra).

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This definition is motivated by the [[Hilbert space]] situation, where an operator $A \in \mathcal{B} (\mathcal{H})$ is called [[positive operator|positive]] if for every vector $x \in \mathcal{H}$ the inequality $\langle x, A x \rangle \ge 0$ holds. If the abstract $C^*$-algebra of the definition above is represented on a Hilbert space, then we see that by [[functional calculus]] we can define a self adjoint operator $B$ by $B \coloneqq f(A)$ with $f(t) := t^{1/2}$ and get $\langle x, A x \rangle = \langle B x, B x \rangle \ge 0$. This shows that the positive elements of the abstract algebra, if represented on a Hilbert space, become positive operators as defined here in the Hilbert space setting.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
A [[linear functional]] $\rho$ on an $C^*$-algebra is \textbf{positive} if $A \ge 0$ implies that $\rho(A) \ge 0$.

A \textbf{state} of a unital $C^*$-algebra is [[linear functional]] $\rho$ such that $\rho$ is positive and $\rho(1) = 1$.

\end{defn}
Though the mathematical notion of state is already close to what physicists have in mind, they usually restrict the set of states further and consider normal states only. We let $\mathcal{R}$ be an $C^*$-algebra and $\pi$ an representation of $\mathcal{R}$ on a Hilbert space $\mathcal{H}$.

\begin{theorem}
\label{}\hypertarget{}{}
A \textbf{normal state} $\rho$ is a state that satisfies one of the following equivalent conditions:

\begin{itemize}%
\item $\rho$ is weak-operator continuous on the unit ball of $\pi(\mathcal{R})$.


\item $\rho$ is strong-operator continuous on the unit ball $\pi(\mathcal{R})$.


\item $\rho$ is ultra-weak continuous.


\item There is an operator $A$ of [[trace class]] of $\mathcal{H}$ with $tr(A) = 1$ such that $\rho(\pi(R)) = tr(A \pi(R))$ for all $R \in \mathcal{R}$.



\end{itemize}
\end{theorem}
This appears as \hyperlink{KadisonRingrose}{KadisonRingrose, def. 7.1.11, theorem 7.1.12}

\begin{remark}
\label{}\hypertarget{}{}
This list is not complete, there are more commonly used equivalent characterizations of normal states.

The last one is most frequently used by physicists, in that context the operator $A$ is also called a [[density matrix]] or density operator.

\end{remark}
Sometimes the observables of a system are described by an abstract $C^*$-algebra, in this case an important notion is the folium:

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{folium} of a representation $\pi$ of an $C^*$-algebra $\mathcal{R}$ on a Hilbert space is the set of normal states of $\pi(\mathcal{R})$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A state $\rho$ of a representation is called a \textbf{vector state} if there is a $x \in \mathcal{H}$ such that $\rho(\pi(R)) = \langle \pi(R)x, x \rangle$ for all $R \in \mathcal{R}$.

\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
Normal states are vector states if $\mathcal{R}$ is a [[von Neumann algebra]] with a separating vector. More precisely: Let $\mathcal{R}$ be a von Neumman algebra acting on a Hilbert space $\mathcal{H}$, let $\rho$ be a normal state of $\mathcal{R}$ and $x \in \mathcal{H}$ be a separating vector for $\mathcal{R}$, then there is a $y \in \mathcal{H}$ such that $\rho(R) = \langle Ry, y \rangle$ for all $R \in \mathcal{R}$.

\end{theorem}
This appears as \hyperlink{KadisonRingrose}{KadisonRingrose, theorem 7.2.3}.

The set of states of an $C^*$-algebra is sometimes called the \textbf{[[space of quantum states|state space]]}.

The state space is non-empty (define a state on the subalgebra $\mathbb{C} 1$ and extend it to the whole $C^*$-algebra via the [[Hahn-Banach theorem]]), convex and weak$^*$-compact, so it has extreme points. By the [[Krein-Milman theorem]] (see Wikipedia: \href{http://en.wikipedia.org/wiki/Krein%E2%80%93Milman_theorem}{Krein-Milman theorem}) it is the weak$^*$-closure of its extreme points.

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{pure state} is a state that is an extreme point of the state space.

\end{defn}
The term ``pure'' originates from the notion of [[entanglement]], a pure state is not a mixture of two distinct other states.

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{prop}
\label{ClassicalProbabilityMeasureAsStateOnMeasurableFunctions}\hypertarget{ClassicalProbabilityMeasureAsStateOnMeasurableFunctions}{}
\textbf{(classical [[probability measure]] as state on [[measurable functions]])}

For $\Omega$ classical [[probability space]], hence a [[measure space]] which normalized total measure $\int_\Omega d\mu = 1$, let $\mathcal{A} \coloneqq L^1(\Omega)$ be the algebra of Lebesgue [[measurable functions]] with values in the [[complex numbers]], regarded as a [[star algebra]] by pointwise [[complex conjugation]]. Then forming the [[expectation value]] with respect to $\mu$ defines a [[state on a star-algebra|state]] (def. \ref{StateOnAStarAlgebra}):

\begin{displaymath}
\itexarray{
    L^1(\Omega) 
      &\overset{\langle (-)\rangle_\mu}{\longrightarrow}& 
    \mathbb{C}
    \\
    A &\mapsto& \int_\Omega A d\mu
  }
\end{displaymath}
\end{prop}
\begin{example}
\label{ElementsOfHilbertSpaceAsPureStates}\hypertarget{ElementsOfHilbertSpaceAsPureStates}{}
\textbf{(elements of a [[Hilbert space]] as [[pure states]] on [[bounded operators]])}

Let $\mathcal{H}$ be a [[complex numbers|complex]] [[separable Hilbert space|separable]] [[Hilbert space]] with [[inner product]] $\langle -,-\rangle$ and let $\mathcal{A} \coloneqq \mathcal{B}(\mathcal{H})$ be the algebra of [[bounded operators]], regarded as a [[star algebra]] under forming [[adjoint operators]]. Then for every element $\psi \in \mathcal{H}$ of unit [[norm]] $\langle \psi,\psi\rangle = 1$ there is the [[state on a star-algebra|state]] (def. \ref{StateOnAStarAlgebra}) given by

\begin{displaymath}
\itexarray{
    \mathcal{B}(\mathcal{H})
      &\overset{\langle (-)\rangle_\psi}{\longrightarrow}&
     \mathbb{C}
     \\
     A &\mapsto& \langle \psi \vert\, A \, \vert \psi \rangle &\coloneqq& \langle \psi, A \psi \rangle
  }
\end{displaymath}
These are [[pure states]] (def. \ref{PureStateOnAStarAlgebra}).

More general states in this case are given by [[density matrices]].

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

\begin{itemize}%
\item [[Fell's theorem]];


\item [[Gleason's theorem]]



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

\begin{itemize}%
\item [[state]], [[quasi-state]]


\item [[quasi-free state]]


\item [[vacuum state]]


\item [[Hadamard state]]


\item [[Alfsen-Shultz theorem]]



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

\hypertarget{exposition}{}\subsubsection*{{Exposition}}\label{exposition}

\begin{itemize}%
\item Jonathan Gleason, \emph{The $C*$-algebraic formalism of quantum mechanics}, 2009 (\href{http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Gleason.pdf}{pdf}, [[GleasonAlgebraic.pdf:file]])

\end{itemize}
\hypertarget{original_articles}{}\subsubsection*{{Original articles}}\label{original_articles}

\begin{itemize}%
\item Richard Kadison, John Ringrose, \emph{Fundamentals of the theory of operator algebras}, AMS (1991)


\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})


\item [[Klaus Fredenhagen]], section 2 of \emph{Algebraische Quantenfeldtheorie}, lecture notes, 2003 ([[FredenhagenAQFT2003.pdf:file]])


\item [[Hans Halvorson]], [[Michael Müger]], def. 1.11 in \emph{Algebraic Quantum Field Theory} (\href{https://arxiv.org/abs/math-ph/0602036}{arXiv:math-ph/0602036})


\item [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], definition 2.4 in \emph{Perturbative algebraic quantum field theory}, In \emph{Mathematical Aspects of Quantum Field Theories}, Springer 2016 (\href{https://arxiv.org/abs/1208.1428}{arXiv:1208.1428})


\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 (\href{https://arxiv.org/abs/1412.5945}{arXiv:1412.5945})


\item [[Katarzyna Rejzner]], section 2.1.2 of \emph{Perturbative Algebraic Quantum Field Theory}, Mathematical Physics Studies, Springer 2016 (\href{https://link.springer.com/book/10.1007%2F978-3-319-25901-7}{web})


\item [[Klaus Fredenhagen]], Falk Lindner, \emph{Construction of KMS States in Perturbative QFT and Renormalized Hamiltonian Dynamics}, Communications in Mathematical Physics Volume 332, Issue 3, pp 895-932, 2014-12-01 (\href{https://arxiv.org/abs/1306.6519}{arXiv:1306.6519})


\item [[Klaas Landsman]], around def. 2.4 in \emph{Foundations of quantum theory -- From classical concepts to Operator algebras}, Springer Open 2017 (\href{https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf}{pdf})


\item [[Michael Dütsch]], section 2.5 of \emph{[[From classical field theory to perturbative quantum field theory]]}, 2018



\end{itemize}
For more references see at \emph{[[operator algebra]]}.

[[!redirects states on a star-algebra]]

[[!redirects states on star-algebras]]

[[!redirects state on an operator algebra]] [[!redirects state of an operator algebra]] [[!redirects states of an operator algebra]] [[!redirects states of operator algebras]] [[!redirects state on an operator algebra]] [[!redirects states on an operator algebra]] [[!redirects states on operator algebras]] [[!redirects state in AQFT]] [[!redirects states in AQFT]] [[!redirects state in operator algebra]] [[!redirects states in operator algebra]] [[!redirects state in AQFT and operator algebra]] [[!redirects states in AQFT and operator algebra]] [[!redirects state of an algebra]] [[!redirects states of an algebra]] [[!redirects states of algebras]] [[!redirects state on an algebra]] [[!redirects states on an algebra]] [[!redirects states on algebras]]

[[!redirects normal state]] [[!redirects normal states]]



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