Friday 26 July 2013

Borchers’ Theorem

Note 2
Borchers’ Theorem  

von Neumann algebra     N
Cyclic and separate vector of N     Ω
Continuous 1 coefficient group of unitary operator     U (λ)   
(λ) has next condition.
(λ)Ω = Ω 
(λ)N U (λ)* N   
Generation operator of (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 (λΔ-it (e-2πtλ)     J U (λJ = (-λ)  

<1 Cyclic vector>
Representation of C*algebra A     {Hπ}
x is called cyclic vector.
<2 separate vector>
Norm space     V
Subset of V     D
sup{||x|| ; xD} < ∞
D is called bounded.
Linear operator from norm space V to norm space V1      T
D ) = V
||Tx||γ (xV )  γ > 0
is called bounded linear operator.
||T || := inf {γ : ||Tx||γ||x|| (xV)} = sup{||Tx|| ; xV, ||x||1} = sup{xV,  x0}
||T || is called norm of T.
Hilbert space     H , K
Bounded linear operator from H  to K     B (HK )
B ( H ) : = B ( HH )
BB (H)
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       U = -∞ eEλ
UeitA = -∞ eitλ Eλ
Continuous 1 coefficient group of unitary operator     {UR}
U= I
Us+t Us Ut   s,t R
Ut* = U-i

To be continued
Tokyo May 2, 2008

No comments:

Post a Comment