Tomita–Takesaki theory

Modular automorphisms of a state

Suppose that M is a von Neumann algebra acting on a Hilbert space H, and Ω is a separating and cyclic 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 φ for the state ϕ ( x ) = ( x Ω , Ω ) of M, so that H is constructed from φ using the GNS construction. We can define an unbounded antilinear operator S₀ on H with domain MΩ by setting

S 0 ( m Ω ) = m ∗ Ω for all m in M, and similarly we can define an unbounded antilinear operator F₀ on H with domain M'Ω by setting F 0 ( m Ω ) = m ∗ Ω 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

S = J | S | = J Δ 1 / 2 = Δ − 1 / 2 J

F = J | F | = J Δ − 1 / 2 = Δ 1 / 2 J

where J = J − 1 = J ∗ is an antilinear isometry called the modular conjugation and Δ = S ∗ S = F S is a positive self adjoint operator called the modular operator.

The main result of Tomita–Takesaki theory states that:

Δ i t M Δ − i t = M

for all t and that

J M J = M ′ ,

the commutant of M.

There is a 1-parameter family of modular automorphisms σ^φ_t of M associated to the state φ, defined by σ ϕ t ( x ) = Δ i t x Δ − i t

The Connes cocycle

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 M for all real t such that

σ ψ t ( x ) = u t σ ϕ t ( x ) u t − 1 so that the modular automorphisms differ by inner automorphisms, and moreover u_t satisfies the 1-cocycle condition

u s + t = u s σ ϕ s ( u t ) 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 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

F ( t ) = ϕ ( A α t ( B ) ) , F ( t + i ) = ϕ ( a t ( B ) A ) ,

Takesaki and Winnink showed that a (faithful semi finite normal) state φ is a KMS state for the 1-parameter group of modular automorphisms σ^φ_−t. 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.)

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 III₀.

A discrete subgroup generated by some x > 0. Then the factor is called a factor of type III_λ with 0 < λ = exp(−2π/x) < 1, or sometimes a Powers factor.

The trivial group 0. Then the factor is called a factor of type III₁. (This is in some sense the generic case.)

Hilbert algebras

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

1. Left multiplication by a fixed a ∈ A is a bounded operator.
2. ♯ is the adjoint; in other words (xy,z) = (y, x^♯z).
3. The involution ^♯ is preclosed
4. 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)*.