Tomita–Takesaki theory

The Tomita - Takesaki theory, named after M. Tomita known and M. Takesaki, also known as modular theory is a theory from the mathematical branch of functional analysis, specifically the theory of von Neumann algebras. A von Neumann algebra is a group of automorphisms associated with the structure of the von Neumann algebra can be investigated.

• 2.1 Cross Products
• 2.2 Type III von Neumann algebras
• 2.3 tensor

Construction

Separating and generating vectors

In a first step, we consider a von Neumann algebra on a Hilbert space for which there is a vector which is both producing and separating for, ie

This is the picture

Well-defined ( since the vector is separating ) and densely defined ( since the vector is causing). From the properties of the involution * follows that conjugate- linear.

Since a vector if and only producing or separating is for, if it is divisive or producing for the commutant, the same situation exists with the same vector for before and you get a close - defined, conjugate- linear map

It can be shown that both operators are lockable. For their financial statements or and. The operator is a composition of two conjugate- linear operators complex - linear, self-adjoint and positive, in general unbounded. The root is called the modular operator, whose existence follows from the Borel calculus for unbounded operators. It also follows that the operators are unitary. It is now the

Set of Tomita: Is the polar decomposition of, then a conjugate- linear isometry with

• And
• For all

By automorphisms on the von Neumann algebra are defined, the mapping is a group homomorphism. Thus, the automorphisms form a group, called the modular group, often the homomorphism is called so.

σ - finite von Neumann algebras

A at the same time producing and separating vector is not always on. The σ - finite von Neumann algebras are precisely those which are isomorphic to those with a generating and separating vector, which are also those who are loyal, have normal conditions, because of this, the desired construct vectors.

Be a faithful, normal state on the von Neumann algebra. Then the GNS construction provides a representation on a Hilbert space and a vector for all. Next is an isomorphism between von Neumann algebras and is a generating and separating vector for. Therefore, one can perform the above presented structure and obtains a modular operator with automorphisms, which can be transferred to by the isomorphism. One thus gets a group homomorphism again

The image or the homomorphism itself is called the modular group associated to. This is a W * - dynamical system.

It raises the question of the dependence of. Can a relationship between automorphism groups and produce and how is determined by? These two questions will be answered next.

KMS condition

We go back by a faithful, normal state on a von Neumann algebra. They say that a group homomorphism satisfies the modular condition regarding if the following applies:

There are two elements with a function:

• Is bounded, continuous and holomorphic
• For everyone.

This condition is also called the KMS condition, named after the physicists Kubo, Martin and Schwinger. One can show that the modular group with respect to meeting the modular condition and is thus even this clearly characterized. This is called a strongly continuous group homomorphism if the pictures are steadily for each with respect to the strong operator topology.

Is a faithful, normal state on a von Neumann algebra, then there exists a strongly continuous group homomorphism which satisfies the modular condition regarding. This is the modular group.

Connes cocycle

We now consider two faithful normal states on the von Neumann algebra. The question, what is the connection between the modular groups and there was answered by Alain Connes as follows:

Are and two faithful normal states on a von Neumann algebra, then there exists a strongly continuous mapping into the unitary group of a von Neumann algebra, so that for the corresponding modular groups and the following applies:

For all and.

Such a map is called a Connes cocycle and the above statement is also known as the Connes cocycle theorem.

General Theory

With somewhat greater technical effort you can also exempt from the requirement of σ - finite. Instead of the normal functional has to look at normal weights and can come to similar conclusions that apply to all von Neumann algebras.

On a von Neumann algebra, there is always faithful, normal and semi- finite weights. Using GNS construction gives a faithful representation on a Hilbert space. Then the conjugate- linear mapping with domain is a tightly - defined lockable operator whose financial statements allows a polar decomposition, so that

• Is a conjugate- linear isometry,
• Is a densely - defined, positive, invertible operator
• For everyone.

Again, we define a homomorphism from the automorphism group of so

This is, again, the modular group and is uniquely determined by a KMS condition is considered in more detail

The modular group is the only strongly -continuous group homomorphism which satisfies the following conditions:

• For all
• Any two elements there is a function with:   is bounded, continuous and holomorphic
• For everyone.

Applications

Cross products

A modular group always defines a W * - dynamical system and one can form the cross product. Since any two such modular groups over a Connes cocycle associated, one can show that the isomorphism class of the cross product are independent of the faithful, normal state. Further, one can show that the cross product thus formed is semi- finite, that is, no type III contains content.

Type III von Neumann algebras

Using the duality properties of W * - dynamical system can be traced back the structure of the type III von Neumann algebras on type II ∞ - algebras. This is known as a set of Takesaki and is described in the article on type III von Neumann algebras.

Tensor

Already Tomita has used this theory to show the so-called commutator set after the commutant of a tensor product of von Neumann algebras is equal to the tensor product of the Kommutanten.