Tomita Takesaki Theory

The Tomita-Takesaki theory , named after M. Tomita and M. Takesaki , also known as modular theory , is a theory from the mathematical branch of functional analysis , more precisely the theory of Von Neumann algebras . A Von Neumann algebra is assigned a group of automorphisms with which the structure of the Von Neumann algebra can be examined more closely.

construction

Separating and generating vectors

In a first step we consider a Von Neumann algebra over a Hilbert space , for which there is a vector that is both generating and separating for , that is ${\displaystyle A}$ ${\displaystyle H}$ ${\displaystyle \xi _{0}\in H}$ ${\displaystyle A}$

${\displaystyle A\xi _{0}:=\{a\xi _{0};\,a\in A\}}$is dense in ( is generating for ) ${\displaystyle H}$ ${\displaystyle \xi _{0}}$ ${\displaystyle A}$

From and follows ( is separating for ) ${\displaystyle a\in A}$ ${\displaystyle a\xi _{0}=0}$ ${\displaystyle a=0}$ ${\displaystyle \xi _{0}}$ ${\displaystyle A}$

That’s the picture

${\displaystyle S_{0}:A\xi _{0}\rightarrow A\xi _{0},\,a\xi _{0}\mapsto a^{*}\xi _{0}}$

well defined (since the vector is separating) and densely defined (since the vector is generating). From the properties of the involution * it follows that is conjugate-linear . ${\displaystyle S_{0}}$

Since a vector is generating or separating for exactly when it is separating or generating for the commutant , the same situation with the same vector also applies to and one obtains a densely-defined, conjugate-linear mapping ${\displaystyle A}$ ${\displaystyle A'}$ ${\displaystyle A'}$

${\displaystyle F_{0}:A'\xi _{0}\rightarrow A'\xi _{0},\,a'\xi _{0}\mapsto a'^{*}\xi _{0}}$.

One can show that both operators are lockable . For their degrees or applies and . As a composition of two conjugate-linear operators, the operator is complex-linear, self-adjoint and positive , in general unbounded. The root is called the modular operator , the existence of which results from the Borel calculus for unbounded operators . This also means that the operators are unitary . The now applies ${\displaystyle S}$ ${\displaystyle F}$ ${\displaystyle S=F_{0}^{*}=F^{*}}$ ${\displaystyle F=S_{0}^{*}=S^{*}}$ ${\displaystyle S^{*}S}$ ${\displaystyle \Delta :=(S^{*}S)^{1/2}=(FS)^{1/2}}$ ${\displaystyle \Delta ^{it},\,t\in \mathbb {R} }$

Tomita theorem  : If the polar decomposition of , then a conjugate-linear isometry with ${\displaystyle S=J\Delta ,\,\Delta =(S^{*}S)^{1/2}}$ ${\displaystyle S}$ ${\displaystyle J:H\rightarrow H}$

• ${\displaystyle J^{2}=\mathrm {id} _{H}}$
• ${\displaystyle JAJ=A'}$ and
• ${\displaystyle \Delta ^{it}A\Delta ^{-it}=A}$ for all ${\displaystyle t\in \mathbb {R} }$

By automorphisms are on the Von Neumann algebra defined, the picture is a group homomorphism . The automorphisms therefore form a group that is called the modular group , often the homomorphism is also called this. ${\displaystyle \sigma _{t}(a):=\Delta ^{it}a\Delta ^{-it}}$ ${\displaystyle \sigma _{t}}$ ${\displaystyle A}$ ${\displaystyle \sigma :\mathbb {R} \rightarrow \mathrm {Aut} (A),t\mapsto \sigma _{t}}$ ${\displaystyle \sigma _{t}}$ ${\displaystyle \sigma }$

σ-finite Von Neumann algebras

A vector that generates and separates at the same time is not always present. The σ-finite Von Neumann algebras are precisely those that are isomorphic to those with a generating and separating vector; these are also those that have faithful, normal states , because the desired vectors can be constructed from these.

Be a faithful, normal state on the Von Neumann algebra . Then the GNS construction provides a representation over a Hilbert space and a vector with for all . Next is an isomorphism between Von Neumann algebras and is a generating and separating vector for . Therefore one can carry out the construction presented above and get a modular operator with automorphisms auf , which can also be transferred to by means of the isomorphism . So you get a group homomorphism again ${\displaystyle \varphi }$ ${\displaystyle A}$ ${\displaystyle \pi _{\varphi }:A\rightarrow L(H_{\varphi })}$ ${\displaystyle H_{\varphi }}$ ${\displaystyle \xi _{\varphi }\in H_{\varphi }}$ ${\displaystyle \varphi (a)=\langle \pi _{\varphi }(a)\xi _{\varphi }|\xi _{\varphi }\rangle }$ ${\displaystyle a\in A}$ ${\displaystyle \pi _{\varphi }:A\rightarrow \pi _{\varphi }(A)}$ ${\displaystyle \xi _{\varphi }}$ ${\displaystyle \pi _{\varphi }(A)}$ ${\displaystyle \Delta _{\varphi }}$ ${\displaystyle \pi _{\varphi }(a)\mapsto \Delta _{\varphi }^{it}\pi _{\varphi }(a)\Delta _{\varphi }^{-it}}$ ${\displaystyle \pi _{\varphi }(A)}$ ${\displaystyle \pi _{\varphi }}$ ${\displaystyle A}$

${\displaystyle \sigma ^{\varphi }:\mathbb {R} \rightarrow \mathrm {Aut} (A),\quad \sigma _{t}^{\varphi }(a):=\pi _{\varphi }^{-1}(\Delta _{\varphi }^{it}\pi _{\varphi }(a)\Delta _{\varphi }^{-it})}$.

The image or the homomorphism itself is called the associated modular group. This is a W * -dynamic system . ${\displaystyle \varphi }$ ${\displaystyle (A,\mathbb {R} ,\sigma ^{\varphi })}$

The question now arises as to the dependence on . You can see a connection between automorphisms groups and produce and how by determined? These two questions will be answered next. ${\displaystyle \varphi }$ ${\displaystyle \sigma ^{\varphi }}$ ${\displaystyle \sigma ^{\psi }}$ ${\displaystyle \sigma ^{\varphi }}$ ${\displaystyle \varphi }$

KMS condition

We assume again a true, normal state on a Von Neumann algebra . One says that a group homomorphism fulfills the modular condition regarding , if the following applies: ${\displaystyle \varphi }$ ${\displaystyle A}$ ${\displaystyle \sigma :\mathbb {R} \rightarrow \mathrm {Aut} (A)}$ ${\displaystyle \varphi }$

For every two elements there is a function with: ${\displaystyle x,y\in A}$ ${\displaystyle f:\{z\in \mathbb {C} |0\leq \mathrm {Im} \,z\leq 1\}\rightarrow \mathbb {C} }$

• ${\displaystyle f}$is bounded, continuous and holomorphic , ${\displaystyle \{z\in \mathbb {C} |0<\mathrm {Im} \,z<1\}}$
• ${\displaystyle f(t)=\varphi (\sigma _{t}(x)y),\quad f(t+i)=\varphi (y\sigma _{t}(x))}$for everyone . ${\displaystyle t\in \mathbb {R} }$

This condition is also called the KMS condition, named after the physicists Kubo , Martin and Schwinger .

It can be shown that the modular group the modular condition related. Fulfilled and that this is thereby even unambiguously characterized. A group homomorphism is called strongly continuous if the mappings for each are continuous with respect to the strong operator topology . ${\displaystyle \sigma ^{\varphi }}$ ${\displaystyle \varphi }$ ${\displaystyle \sigma :\mathbb {R} \rightarrow \mathrm {Aut} (A)}$ ${\displaystyle t\mapsto \sigma _{t}(a)}$ ${\displaystyle a\in A}$

If there is a true, normal state on a Von Neumann algebra , then there is exactly one strongly continuous group homomorphism that fulfills the modular condition with respect to . This is the modular group . ${\displaystyle \varphi }$ ${\displaystyle A}$ ${\displaystyle \sigma :\mathbb {R} \rightarrow \mathrm {Aut} (A)}$ ${\displaystyle \varphi }$ ${\displaystyle \sigma ^{\varphi }}$

Connes-Kozykel

We now consider two faithful, normal states on the Von Neumann algebra . The question of what is the connection between the modular groups and there was of Alain Connes answered as follows: ${\displaystyle A}$ ${\displaystyle \sigma ^{\varphi }}$ ${\displaystyle \sigma ^{\psi }}$

If and are two true, normal states on a Von Neumann algebra , there is a strongly continuous mapping into the unitary group of the Von Neumann algebra, so that for the associated modular groups and the following applies: ${\displaystyle \varphi }$ ${\displaystyle \psi }$ ${\displaystyle A}$ ${\displaystyle u:\mathbb {R} \rightarrow U(A)}$ ${\displaystyle \sigma ^{\varphi }}$ ${\displaystyle \sigma ^{\psi }}$

• ${\displaystyle \sigma _{t}^{\psi }(a)=u_{t}\sigma _{t}^{\varphi }(a)u_{t}^{*}}$
• ${\displaystyle u_{t+s}=u_{t}\sigma _{t}^{\varphi }(u_{s})}$

for everyone and . ${\displaystyle s,t\in \mathbb {R} }$ ${\displaystyle a\in A}$

Such a mapping is called a Connes-Kozykel and the above statement is also known as the Connes-Kozykel theorem. ${\displaystyle u:\mathbb {R} \rightarrow U(A)}$

General theory

With a little more technical effort, one can also free oneself from the requirement of σ-finiteness. Instead of normal functionals, one must consider normal weights and very similar results can be obtained, which apply to all Von Neumann algebras.

On a Von Neumann algebra there are always faithful, normal and semi-finite weights . A true representation on a Hilbert space is obtained by means of a GNS construction . Then the conjugate-linear mapping with domain is a densely-defined closable operator on , the termination of which allows a polar decomposition , so that ${\displaystyle A}$ ${\displaystyle \omega }$ ${\displaystyle \pi _{\omega }:A\rightarrow L(H_{\omega })}$ ${\displaystyle H_{\omega }}$ ${\displaystyle x\mapsto x^{*}}$ ${\displaystyle \{x\in A;\omega (x^{*}x)<\infty ,\omega (xx^{*})<\infty \}\subset H_{\omega }}$ ${\displaystyle H_{\omega }}$ ${\displaystyle S}$ ${\displaystyle S=J\Delta }$

• ${\displaystyle J}$ is a conjugate-linear isometry, ${\displaystyle J^{2}=\mathrm {id} _{H_{\omega }}}$
• ${\displaystyle \Delta }$ is a densely-defined, positive, invertible operator
• ${\displaystyle J\pi _{\omega }(A)J=\pi _{\omega }(A)'}$
• ${\displaystyle \Delta ^{it}\pi _{\omega }(A)\Delta ^{-it}=\pi _{\omega }(A)}$for everyone . ${\displaystyle t\in \mathbb {R} }$

Again a homomorphism of is defined in the automorphism group of such that ${\displaystyle \sigma }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle A}$

${\displaystyle \pi _{\omega }(\sigma _{t}(a))=\Delta ^{it}\pi _{\omega }(a)\Delta ^{-it}}$for everyone . ${\displaystyle a\in A}$

This is again called the modular group and is clearly defined by a KMS condition, more precisely applies

The modular group is the only strongly continuous group homomorphism that satisfies the following conditions: ${\displaystyle \sigma :\mathbb {R} \rightarrow \mathrm {Aut} (A)}$

• ${\displaystyle \omega (\sigma _{t}(a))=\omega (a)}$ for all ${\displaystyle A\in A^{+},t\in \mathbb {R} }$
• for every two elements there is a function with: ${\displaystyle a,b\in \{x\in A;\omega (x^{*}x)<\infty ,\omega (xx^{*})<\infty \}}$ ${\displaystyle f:\{z\in \mathbb {C} |0\leq \mathrm {Im} \,z\leq 1\}\rightarrow \mathbb {C} }$
  • ${\displaystyle f}$is bounded, continuous and holomorphic, ${\displaystyle \{z\in \mathbb {C} |0<\mathrm {Im} \,z<1\}}$
  • ${\displaystyle f(t)=\omega (\sigma _{t}(a)b),\quad f(t+i)=\omega (b\sigma _{t}(a))}$for everyone . ${\displaystyle t\in \mathbb {R} }$

Applications

Cross products

A modular group always defines a W * -dynamic system and the cross product can be formed. Since two such modular groups are connected via a Connes cocycle, it can be shown that the isomorphism class of the cross product does not depend on the selected, faithful, normal state. It can also be shown that the cross product formed in this way is semi-sterile , that is to say does not contain any type III content. ${\displaystyle \sigma :\mathbb {R} \rightarrow \mathrm {Aut} (A)}$ ${\displaystyle (A,\mathbb {R} ,\sigma )}$ ${\displaystyle A\ltimes _{\sigma }\mathbb {R} }$

Type III Von Neumann algebras

Using the duality properties of the W * -dynamic system, the structure of the type III Von Neumann algebras can be reduced to type II _∞ algebras . This is known as the Takesaki Theorem and is described in the article on Type III Von Neumann Algebras .

Tensor products

Tomita already used this theory to show the so-called commutator theorem, according to which the commutant of a tensor product of Von Neumann algebras is equal to the tensor product of the commutants.

Individual evidence

1. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Theorem 9.2.9
2. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Theorem 9.2.16
3. ↑ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , Theorem 5.5.11
4. ^ A. van Daele: Continuous crossed products and type III von Neumann algebras , Cambridge University Press (1978), ISBN 0-521-21975-2 , Theorem II 2.2
5. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , A further extension of modular theory , from page 639
6. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , A further extension of modular theory , Theorem 9.2.37
7. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , A further extension of modular theory , Theorem 9.2.38
8. ^ A. van Daele: Continuous crossed products and type III von Neumann algebras , Cambridge University Press (1978), ISBN 0-521-21975-2 , part II, paragraph 3
9. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , Theorem 11.2.16