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{{Short description|Mathematical method in functional analysis}}
In the theory of [[von Neumann algebra]]s, a part of the mathematical field of [[functional analysis]], '''Tomita–Takesaki theory'''  is a method for constructing '''modular automorphisms''' of von Neumann algebras from the [[polar decomposition]] of a certain involution. It is essential for the theory of [[von Neumann algebra#Type III factors|type III factors]], and has led to a good structure theory for these previously intractable objects.

The theory was introduced by {{harvs|txt|first=Minoru |last=Tomita|authorlink=Minoru Tomita|year=1967}}, but his work was hard to follow and mostly unpublished, and little notice was taken of it until {{harvs|txt|first=Masamichi |last=Takesaki|year=1970|authorlink=Masamichi Takesaki}} wrote an account of Tomita's theory.<ref>{{harvnb|Takesaki|2003|pages=38–39}}</ref>

==Modular automorphisms of a state==

Suppose that ''M'' is a von Neumann algebra acting on a [[Hilbert space]] ''H'', and Ω is a [[cyclic and separating vector]] of ''H'' of norm 1. ('''Cyclic''' means that ''MΩ'' is dense in ''H'', and '''separating''' means that the map from ''M'' to ''MΩ'' is injective.) We write <math>\phi</math> for the vector state <math>\phi(x) = (x\Omega, \Omega)</math> of ''M'', so that ''H'' is constructed from <math>\phi</math> using the [[Gelfand–Naimark–Segal construction]]. Since Ω is separating, <math>\phi</math> is faithful.

We can define a (not necessarily bounded) antilinear operator ''S''<sub>0</sub> on ''H'' with dense domain ''MΩ'' by setting <math>S_0(m\Omega) = m^*\Omega</math> for all ''m'' in ''M'', and similarly we can define a (not necessarily bounded) antilinear operator ''F''<sub>0</sub> on ''H'' with dense domain ''M'Ω'' by setting <math>F_0(m\Omega) = m^*\Omega</math> for ''m'' in ''M''&prime;, where ''M''&prime; is the [[commutant]] of ''M''.

These operators are closable, and we denote their closures by ''S'' and ''F'' = ''S''*. They have [[polar decomposition]]s

: <math>\begin{align}
  S = J|S| &= J\Delta^\frac{1}{2} = \Delta^{-\frac{1}{2}}J \\
  F = J|F| &= J\Delta^{-\frac{1}{2}} = \Delta^\frac{1}{2}J
\end{align}</math>

where <math>J = J^{-1} = J^*</math> is an antilinear isometry of ''H'' called the ''modular conjugation'' and <math>\Delta = S^*S = FS</math> is a positive (hence, self-adjoint) and densely defined operator called the ''modular operator''.

=== Commutation theorem ===
The main result of Tomita–Takesaki theory states that:
: <math>\Delta^{it}M\Delta^{-it} = M</math>

for all ''t'' and that
: <math>JMJ = M',</math>

the commutant of ''M''.

There is a 1-parameter group of ''modular automorphisms'' <math>\sigma^{\phi_t}</math> of ''M'' associated with the state <math>\phi</math>, defined by <math>\sigma^{\phi_t}(x) = \Delta^{it}x\Delta^{-it}</math>.

The modular conjugation operator ''J'' and the 1-parameter unitary group <math>\Delta^{it}</math> satisfy

:<math>J\Delta^{it}J =\Delta^{it}</math>

and

:<math>J\Delta J = \Delta^{-1}.</math>

==The Connes cocycle==

The modular automorphism group of a von Neumann algebra ''M'' depends on the choice of state φ. [[Alain Connes|Connes]] discovered that changing the state does not change the image of the modular automorphism in the [[outer automorphism group]] of ''M''. More precisely, given two faithful states φ and ψ of ''M'', we can find unitary elements ''u<sub>t</sub>'' of ''M'' for all real ''t'' such that

: <math>\sigma^{\psi_t}(x) = u_t\sigma^{\phi_t}(x)u_t^{-1} </math> 
so that the modular automorphisms differ by inner automorphisms, and moreover ''u<sub>t</sub>'' satisfies the 1-cocycle condition

: <math>u_{s+t} = u_s\sigma^{\phi_s}(u_t)</math>
In particular, there is a canonical homomorphism from the additive group of reals to the outer automorphism group of ''M'', that is independent of the choice of faithful state.

==KMS states==
The term ''KMS state'' comes from the Kubo–Martin–Schwinger condition in [[quantum statistical mechanics]].

A '''[[KMS state]]''' <math>\phi</math> on a von Neumann algebra ''M'' with a given 1-parameter group of automorphisms α<sub>''t''</sub> is a state fixed by the automorphisms  such that for every pair of elements ''A'', ''B'' of ''M'' there is a bounded continuous function ''F'' in the strip {{nowrap|0 ≤ Im(''t'') ≤ 1}}, holomorphic in the interior, such that

: <math>\begin{align}
      F(t) &= \phi(A\alpha_t(B)),\\
  F(t + i) &= \phi(\alpha_t(B)A)
\end{align}</math>

Takesaki and Winnink showed that any (faithful semi finite normal) state <math>\phi</math> is a KMS state for the 1-parameter group of modular automorphisms <math>\sigma^{\phi_{-t}}</math>. Moreover, this characterizes the modular automorphisms of <math>\phi</math>.

(There is often an extra parameter, denoted by β, used in the theory of KMS states. In the description above this has been  normalized to be 1 by rescaling the 1-parameter family of automorphisms.)

==Structure of type III factors==

We have seen above that there is a canonical homomorphism δ from the group of reals to the outer automorphism group of a von Neumann algebra, given by modular automorphisms.  The kernel of δ is an important invariant of the algebra. For simplicity assume that the von Neumann algebra is a factor. Then the possibilities for the kernel  of δ are:

* The whole real line. In this case δ is trivial and the factor is type I or II.
* A proper dense subgroup of the real line. Then the factor is called a factor of type III<sub>0</sub>.
* A discrete subgroup generated by some ''x''&nbsp;>&nbsp;0. Then the factor is called a factor of type III<sub>λ</sub> with 0&nbsp;<&nbsp;λ&nbsp;=&nbsp;exp(−2''π''/''x'')&nbsp;<&nbsp;1, or sometimes a Powers factor.
* The trivial group 0. Then the factor is called a factor of type III<sub>1</sub>. (This is in some sense the generic case.)

== Left Hilbert algebras ==
{{see also|Commutation theorem for traces#Hilbert algebras}}

The main results of Tomita–Takesaki theory were proved using left and right Hilbert algebras.<ref>{{harvnb|Takesaki|2003|pages=1–39}}</ref>

A '''left Hilbert algebra''' is an algebra <math>\mathfrak A</math> with involution ''x'' &rarr; ''x''<sup>♯</sup> and an inner product (·,·) such that
# Left multiplication by a fixed ''a'' &isin; <math>\mathfrak A</math> is a bounded operator.
# ♯ is the adjoint; in other words {{nowrap|(''xy'', ''z'') {{=}} (''y'', ''x''<sup>♯</sup>''z'')}}.
#The involution <sup>♯</sup> is preclosed.
# The subalgebra spanned by all products ''xy'' is dense in <math>\mathfrak |</math> w.r.t. the inner product.

A '''right Hilbert algebra''' is defined similarly (with an involution ♭) with left and right reversed in the conditions above.

A (unimodular) '''Hilbert algebra''' is a left Hilbert algebra for which ♯ is an isometry, in other words {{nowrap|(''x'', ''y'') {{=}} (''y''<sup>♯</sup>, ''x''<sup>♯</sup>)}}. In this case the involution is denoted by ''x''* instead of ''x''<sup>♯</sup> and coincides with ''modular conjugation'' ''J''. This is the special case of [[Commutation theorem for traces#Hilbert algebras|Hilbert algebras]]. The ''modular operator'' is trivial and the corresponding von Neumann algebra is a direct sum of type I and type II von Neumann algebras.

Examples:
* If ''M'' is a von Neumann algebra acting on a Hilbert space ''H'' with a cyclic separating unit vector ''v'', then put {{nowrap|<math>\mathfrak A</math> {{=}} ''Mv''}} and define {{nowrap|(''xv'')(''yv'') {{=}} ''{{text|xyv}}''}} and {{nowrap|(''xv'')<sup>♯</sup> {{=}} ''x''*''v''}}.  The vector ''v'' is the identity of <math>\mathfrak A</math>, so <math>\mathfrak A</math> is a unital left Hilbert algebra.<ref name="tak-2">{{harvnb|Takesaki|2003|page=2}}</ref>
* If ''G'' is a locally compact group, then the vector space of all continuous complex functions on ''G'' with compact support is a right Hilbert algebra if multiplication is given by convolution, and {{nowrap|''x''<sup>♭</sup>(''g'')  {{=}} ''x''(''g''<sup>−1</sup>)*}}.<ref name="tak-2"/>

For a fixed left Hilbert algebra <math>\mathfrak A</math>, let ''H'' be its Hilbert space completion. Left multiplication by ''x'' yields a bounded operator λ(''x'') on ''H'' and hence a *-homomorphism λ of <math>\mathfrak A</math> into ''B''(''H''). The *-algebra <math>\lambda({\mathfrak A})</math> generates the von Neumann algebra

:<math>{\cal R}_\lambda({\mathfrak A}) = \lambda({\mathfrak A})^{\prime\prime}.</math>

Tomita's key discovery concerned the remarkable properties of the closure of the operator <sup>♯</sup> and its polar decomposition. If ''S'' denotes this closure (a conjugate-linear unbounded operator), let Δ = ''S''* ''S'', a positive unbounded operator. Let ''S'' = ''J'' Δ<sup>1/2</sup> denote its [[polar decomposition]]. Then ''J'' is a conjugate-linear isometry satisfying<ref>{{harvnb|Takesaki|2003|page=4}}</ref>

:<math>S=S^{-1},\,\,\,</math> <math>J^2 =I,\,\,\,</math> <math>J\Delta J =\Delta^{-1}\,\,\,</math>and <math>\, S=\Delta^{-1/2} J</math>.

Δ is called the '''modular operator''' and ''J'' the '''modular conjugation'''.

In {{harvtxt|Takesaki|2003|pages=5–17}}, there is a self-contained proof of the main '''commutation theorem''' of Tomita-Takesaki:

:<math>\Delta^{it}{\cal R}_\lambda({\mathfrak A})\Delta^{-it} =  {\cal R}_\lambda({\mathfrak A})\,\,</math> and <math>\,\,J{\cal R}_\lambda({\mathfrak A})J = {\cal R}_\lambda({\mathfrak A})^{\prime}.</math>

The proof hinges on evaluating the operator integral:<ref>{{harvnb|Takesaki|2003|pages=15–16}}</ref>

:<math>e^{s/2} \Delta^{1/2}\,(\Delta + e^s)^{-1} =  \int_{-\infty}^\infty {e^{-ist}\over e^{\pi t} + e^{-\pi t}} \,\Delta^{it} \, {\rm d} t.</math>

By the [[spectral theorem]],{{sfn|Rudin|1991}} that is equivalent to proving the equality with ''e''<sup>''x''</sup> replacing Δ; the identity for scalars follows by contour integration. It reflects the well-known fact that, with a suitable normalisation, the function <math>{\rm sech}</math> is its own Fourier transform.

== Notes ==
{{reflist|30em}}

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