Tomita–Takesaki theory

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. 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. The theory was introduced by Minoru Tomita (1967), but his work was hard to follow and mostly unpublished, and little notice was taken of it until Masamichi Takesaki (1970) wrote an account of Tomita's theory. 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 {displaystyle phi } for the state ϕ ( x ) = ( x Ω , Ω ) {displaystyle phi (x)=(xOmega ,Omega )} of M, so that H is constructed from ϕ {displaystyle phi } 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 {displaystyle S_{0}(mOmega )=m^{*}Omega } 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 {displaystyle F_{0}(mOmega )=m^{*}Omega } 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 J = J − 1 = J ∗ {displaystyle J=J^{-1}=J^{*}} is an antilinear isometry called the modular conjugation and Δ = S ∗ S = F S {displaystyle Delta =S^{*}S=FS} is a positive self adjoint operator called the modular operator.