In the theory of von Neumann algebras, 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 type III factors, and has led to a good structure theory for these previously intractable objects.

- Modular automorphisms of a state
- The Connes cocycle
- KMS states
- Structure of type III factors
- Hilbert algebras
- References

The theory was introduced by MinoruTomita ( 1967 ), but his work was hard to follow and mostly unpublished, and little notice was taken of it until MasamichiTakesaki ( 1970 ) wrote an account of Tomita's theory.

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 for the state of *M*, so that *H* is constructed from using the Gelfand–Naimark–Segal construction.

We can define an unbounded antilinear operator *S*_{0} on *H* with domain *MΩ* by setting for all *m* in *M*, and similarly we can define an unbounded antilinear operator *F*_{0} on *H* with domain *M'Ω* by setting for *m* in *M*′, where *M*′ is the commutant of *M*.

These operators are closable, and we denote their closures by *S* and *F* = *S**. They have polar decompositions

where is an antilinear isometry called the modular conjugation and is a positive self-adjoint operator called the *modular operator*.

The main result of Tomita–Takesaki theory states that:

for all *t* and that

the commutant of *M*.

There is a 1-parameter family of **modular automorphisms** of *M* associated to the state , defined by

The modular automorphism group of a von Neumann algebra *M* depends on the choice of state φ. 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 _{t}* of

so that the modular automorphisms differ by inner automorphisms, and moreover *u _{t}* satisfies the 1-cocycle condition

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.

The term *KMS state* comes from the Kubo–Martin–Schwinger condition in quantum statistical mechanics.

A ** KMS state ** φ on a von Neumann algebra *M* with a given 1-parameter group of automorphisms α_{t} 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 0 ≤ Im(*t*) ≤ 1, holomorphic in the interior, such that

Takesaki and Winnink showed that a (faithful semi finite normal) state φ is a KMS state for the 1-parameter group of modular automorphisms . Moreover, this characterizes the modular automorphisms of φ.

(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.)

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
_{0}. - A discrete subgroup generated by some
*x*> 0. Then the factor is called a factor of type III_{λ}with 0 < λ = exp(−2*π*/*x*) < 1, or sometimes a Powers factor. - The trivial group 0. Then the factor is called a factor of type III
_{1}. (This is in some sense the generic case.)

The main results of Tomita–Takesaki theory were proved using left and right Hilbert algebras.

A **left Hilbert algebra** is an algebra with involution *x*→*x*^{♯} and an inner product (,) such that

- Left multiplication by a fixed
*a*∈*A*is a bounded operator. - ♯ is the adjoint; in other words (
*xy*,*z*) = (*y*,*x*^{♯}*z*). - The involution
^{♯}is preclosed - The subalgebra spanned by all products
*xy*is dense in*A*.

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

A **Hilbert algebra** is a left Hilbert algebra such that in addition ♯ is an isometry, in other words (*x*, *y*) = (*y*^{♯}, *x*^{♯}).

Examples:

- If
*M*is a von Neumann algebra acting on a Hilbert space*H*with a cyclic separating vector*v*, then put*A*=*Mv*and define (*xv*)(*yv*) =*xyv*and (*xv*)^{♯}=*x***v*. Tomita's key discovery was that this makes*A*into a left Hilbert algebra, so in particular the closure of the operator^{♯}has a polar decomposition as above. The vector*v*is the identity of*A*, so*A*is a unital left Hilbert algebra. - 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*x*^{♭}(*g*) =*x*(*g*^{−1})*.

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