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.
The theory was introduced by Minoru Tomita (1967), but his work was hard to follow and mostly unpublished, and little notice was taken of it until Masamichi Takesaki (1970) wrote an account of Tomita's theory.
[1] 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.
We can define a (not necessarily bounded) antilinear operator S0 on H with dense domain MΩ by setting
for all m in M, and similarly we can define a (not necessarily bounded) antilinear operator F0 on H with dense 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*.
The main result of Tomita–Takesaki theory states that: for all t and that the commutant of M. There is a 1-parameter group of modular automorphisms
The modular conjugation operator J and the 1-parameter unitary group
satisfy and 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 ut of M for all real t such that so that the modular automorphisms differ by inner automorphisms, and moreover ut 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.
on a von Neumann algebra M with a given 1-parameter group of automorphisms αt 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 any (faithful semi finite normal) state
is a KMS state for the 1-parameter group of modular automorphisms
(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 main results of Tomita–Takesaki theory were proved using left and right Hilbert algebras.
In this case the involution is denoted by x* instead of x♯ and coincides with modular conjugation J.
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: For a fixed left Hilbert algebra
generates the von Neumann algebra Tomita's key discovery concerned the remarkable properties of the closure of the operator ♯ and its polar decomposition.
5–17), there is a self-contained proof of the main commutation theorem of Tomita-Takesaki: The proof hinges on evaluating the operator integral:[5] By the spectral theorem,[6] that is equivalent to proving the equality with ex replacing Δ; the identity for scalars follows by contour integration.
It reflects the well-known fact that, with a suitable normalisation, the function