Friday 26 July 2013

Tomita’s Fundamental Theorem

Note 1
Tomita’s Fundamental Theorem  

(1) JNJ = 
(2) ΔitNΔ-it = N,  R

<1 von Neumann Algebra>
von Neumann algebra     *subalgebra satisfies A ’’ = A
<1-1 *subalgebra> 
Algebra that has involution*       *algebra
Element of *algebra     AA
When A = A*, A is called self-adjoint.
When A *AAA*, A is called normal.
When A A*= 1, A is called unitary.
Subset of     B
* := B*B
When 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
Arbitrary ABAλC
(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)itR.
Δis unbounded positive self-adjoint operator.
Δit is 1 coefficient unitary group.
<2-2 Modular conjugation>
J is adjoint linear isometric operator, JI.
<2-3 Symmetric operator>
Objection operator from HR to KiK     PQ
R = P + Q
Polar decomposition of P - Q at HR     P – Q = JT
T is positive symmetric operator over HR.
Re<xTy> = Re<Txy>
T2 = (P – Q)2
<2-4 Polar decomposition>
Partial isometric operator     VN
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 = ’>
<For more details>

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