Note 2
Borchers’ Theorem
[Theorem]
von Neumann algebra N
Cyclic and separate vector of N Ω
Continuous 1 coefficient group of unitary operator U (λ)
U (λ) has next condition.
U (λ)Ω = Ω
U (λ)N U (λ)* ⊂N
Generation operator of U (λ) H
Modular operator on (N, Ω) Δ
Modular conjugation on (N, Ω) J
Next 2 conditions are equivalent.
(i) H ≧ 0
(ii) Δit U (λ) Δ-it = U (e-2πtλ) J U (λ) J = U (-λ)
[Preparation]
<1 Cyclic vector>
Representation of C*algebra A {H, π}
x∈H
{π(A)x} - = H
x is called cyclic vector.
<2 separate vector>
Norm space V
Subset of V D
sup{||x|| ; x∈D} < ∞
D is called bounded.
Linear operator from norm space V to norm space V1 T
D ( T ) = V
||Tx||≦γ (x∈V ) γ > 0
T is called bounded linear operator.
||T || := inf {γ : ||Tx||≦γ||x|| (x∈V)} = sup{||Tx|| ; x∈V, ||x||≦1} = sup{; x∈V, x≠0}
||T || is called norm of T.
Hilbert space H , K
Bounded linear operator from H to K B (H, K )
B ( H ) : = B ( H, H )
B⊂B (H)
x∈H
Q⊂B
Qx = 0 → Q = 0
x is called separate vector.
<3 Continuous 1 coefficient group of unitary operator >
Self-adjoint operator A
Spectrum measure {Eλ}
A = ∫∞-∞ λdEλ
Unitary operator over H U = ∫∞-∞ eiλEλ
Ut = eitA = ∫∞-∞ eitλ Eλ
Continuous 1 coefficient group of unitary operator {Ut ; t ∈R}
U0 = I
Us+t = Us + Ut ∀s,t ∈R
Ut* = U-i
<4 Spectrum Measure>
To be continued
Tokyo May 2, 2008
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