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\begin{document}

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\section*{von Neumann algebra factor}

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

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

[[!include higher algebra - contents]]

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

[[!include functional analysis - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{classification}{Classification}\dotfill \pageref*{classification} \linebreak
\noindent\hyperlink{type_i}{Type I}\dotfill \pageref*{type_i} \linebreak
\noindent\hyperlink{type_ii}{Type II}\dotfill \pageref*{type_ii} \linebreak
\noindent\hyperlink{type_iii}{Type III}\dotfill \pageref*{type_iii} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{subfactors}{Subfactors}\dotfill \pageref*{subfactors} \linebreak
\noindent\hyperlink{relation_to_quantum_field_theory}{Relation to quantum field theory}\dotfill \pageref*{relation_to_quantum_field_theory} \linebreak


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

For $A$ a [[von Neumann algebra]] write $A'$ for its [[commutant]] in the ambient algebra $B(\mathcal{H})$ of [[bounded operator]]s.

\begin{defn}
\label{}\hypertarget{}{}
A von Neumann algebra $A$ is called a \textbf{factor} if its [[center]] is trivial

\begin{displaymath}
Z(A) := A \cap A' = \mathbb{C}1
  \,.
\end{displaymath}
Equivalently: if $A$ and its [[commutant]] $A'$ generate the full algebra of [[bounded operators]] $B(\mathcal{H})$.

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

Every von Neumann algebra may be written as a [[direct integral]] over factors. (\hyperlink{vNeumann49}{von Neumann 49})

\hypertarget{classification}{}\subsection*{{Classification}}\label{classification}

Factors are classified in terms of the [[K-theory]] of their categories of finite [[W*-modules]]. A [[W*-module]] over a factor $A$ is \emph{finite} if it is not isomorphic to its proper submodule.

\hypertarget{type_i}{}\subsubsection*{{Type I}}\label{type_i}

Type I factors are characterized by the condition that the K-theory of finite modules is isomorphic to $\mathbf{Z}$, the group of integers. The only factors of this type are of the from $B(H)$, bounded operators on a Hilbert spacre{\tt \symbol{126}}$H$.

\hypertarget{type_ii}{}\subsubsection*{{Type II}}\label{type_ii}

Type II factors are characterized by the condition that the K-theory of finite modules is isomorphic to $\mathbf{R}$, the group of real numbers.

Type II factors are subdivided into two classes: type II$_1$ factors are characterized by the condition that $A$ is a finite $A$-module, whereas for a type II$_\infty$ factor $A$ is not a finite $A$-module.

\hypertarget{type_iii}{}\subsubsection*{{Type III}}\label{type_iii}

Type III factors are characterized by the condition that the K-theory of finite modules is trivial, i.e., only the zero module is finite.

Type III factors are further subdivided into three classes, according to the structure of the center of their [[modular algebra]], which is a commutative von Neumann algebra graded by purely imaginary numbers, whose graded components are [[noncommutative L{\tt \symbol{94}}p-spaces]].

By the [[von Neumann duality]] for commutative von Neumann algebras, the spectrum of this center is a [[measurable space]] equipped with a σ-ideal of negligible sets and the grading yields an action of $\mathbf{R}$, the group of real numbers. This object is known as the [[noncommutative flow of weights]].

If the center is trivial (so the spectrum is a point), the factor has type III$_1$. If the action of $\mathbf{R}$ is not periodic, then the factor has type III$_0$. If the action is periodic with period $\lambda$, a positive real number, then the factor has type III$_{\exp(-\lambda)}$.

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

\begin{itemize}%
\item [[C-star algebra]]


\item [[uniformly hyperfinite algebra]]



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

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

The original sources are

\begin{itemize}%
\item Murray, [[John von Neumann]], \ldots{}


\item [[John von Neumann]], \emph{On rings of operators, reduction theory}, Annals of Mathematics Second Series, Vol. 50, No. 2 (1949) (\href{http://www.jstor.org/stable/1969463}{jstor})



\end{itemize}
\begin{itemize}%
\item [[Alain Connes]], \ldots{}

\end{itemize}
Lecture notes include

\begin{itemize}%
\item V.S. Sunder, \emph{von Neumann algebras, $II_1$-factors, and their subfactors} (\href{http://www.math.iitb.ac.in/seminar/archives/sunder_iitb3.pdf}{pdf})


\item Hideki Kosaki, \emph{Type III factors and index theory} (1993) (\href{http://pages.uoregon.edu/njp/lec-f.pdf}{pdf})



\end{itemize}
\hypertarget{subfactors}{}\subsubsection*{{Subfactors}}\label{subfactors}

The mathematics of inclusions of \emph{subfactors} is giving deep structural insights. See also at \emph{[[planar algebra]]}.

\begin{itemize}%
\item [[Vaughan Jones]],

\emph{Index for subfactors}, Invent. Math. \textbf{72}, I (I983);

\emph{A polynomial invariant for links via von Neumann algebras}, Bull. AMS \textbf{12}, 103 (1985);

\emph{Hecke algebra representations of braid groups and link polynomials}, Ann. Math. \textbf{126}, 335 (1987)


\item [[Vaughan Jones]], [[Scott Morrison]], [[Noah Snyder]], \emph{The classification of subfactors of index at most 5} (\href{http://arxiv.org/abs/1304.6141}{arXiv:1304.6141})


\item Vaughan F. R. Jones, David Penneys, \emph{Infinite index subfactors and the GICAR categories}, \href{http://arxiv.org/abs/1410.0856}{arxiv/1410.0856}



\end{itemize}
\hypertarget{relation_to_quantum_field_theory}{}\subsubsection*{{Relation to quantum field theory}}\label{relation_to_quantum_field_theory}

\begin{itemize}%
\item [[Jakob Yngvason]], \emph{The Role of Type III Factors in Quantum Field Theory} (\href{https://arxiv.org/abs/math-ph/0411058}{arXiv:math-ph/0411058})

\end{itemize}
[[!redirects von Neumann algebra factors]]

[[!redirects von Neumann algebra subfactor]] [[!redirects von Neumann algebra subfactors]]

[[!redirects subfactor]] [[!redirects subfactors]]



\end{document}