Note 1
Tomita’s Fundamental Theorem
[Theorem]
(1) JNJ = N ’
(2) ΔitNΔ-it = N, ∀t ∈R
[Preparation]
<1 von Neumann Algebra>
von Neumann algebra *subalgebra satisfies A ’’ = A.
<1-1 *subalgebra>
Algebra A that has involution* *algebra
Element of *algebra A∈A
When A = A*, A is called self-adjoint.
When A *A= AA*, A is called normal.
When A A*= 1, A is called unitary.
Subset of A B
B * := B*∈B
When B = B*, B is called self-adjoint set.
Subalgebra of A B
When B is adjoint set, B is called *subalgebra.
<1-2 involution*>
Involution * over algebra A over C is map * that satisfies next condition.
Map * : A∈A ↦ A*∈A
Arbitrary A, B∈A, λ∈C
(i) (A*)* = A
(ii) (A+B)* = A*+B*
(iii) (λA)* =λ-A*
(iiii) (AB)* = B*A*
<2 Modular operator, modular conjugation>
Δ is called modular operator on x0.
J is called modular conjugation on x0.
<2-1 Modular operator>
Δ = R-1(2I-R), Δit = R-it(2I-R)it, t ∈R.
Δis unbounded positive self-adjoint operator.
Δit is 1 coefficient unitary group.
<2-2 Modular conjugation>
J is adjoint linear isometric operator, J2 = I.
<2-3 Symmetric operator>
Objection operator from HR to K, iK P, Q
R = P + Q
Polar decomposition of P - Q at HR P – Q = JT
T is positive symmetric operator over HR.
Re<x, Ty> = Re<Tx, y>
T2 = (P – Q)2
<2-4 Polar decomposition>
φ∈N*, ψ∈N*,+
Partial isometric operator V∈N
φ = RVψ and V*V = s(ψ)
|| φ|| = || ψ ||
ψ is called absolute value ofφ.
φ = RVψ is called polar decomposition ofφ.
<2-5 N* >
Bounded linear functional over N N*
To be continued
Tokyo May 3, 2008
[Postscript August 2, 2008]
<On [Theorem] (1) JNJ = N ’>
<For more details>
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