von Neumann Algebra 4

Note 2

Borchers’ Theorem

TANAKA Akio

[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) Δ^it 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 {E_λ}

A = ∫^∞_-∞ λdE_λ

Unitary operator over H U = ∫^∞_-∞ e^iλE_λ

U_t = e^itA = ∫^∞_-∞ e^itλ E_λ

Continuous 1 coefficient group of unitary operator {U_t ; t ∈R}

U₀ = I

U_s+t = U_s + U_t ∀s,t ∈R

U_t* = U_-i

<4 Spectrum Measure>

To be continued

Tokyo May 2, 2008

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