von Neumann Algebra 3 Note 2 Purely Infinite

 

von Neumann Algebra 3

Note 2

Purely Infinite  

TANAKA Akio

[Theorem]

The necessary and sufficient condition for what von Neumann algebra N is purely infinite ( Ⅲtype) is what semi-finite normal trace that is not 0 does not exist over N.

[Explanation]

<1 Trace>

<1-1>

Trace over von Neumann algebra N          τ : N₊ → [0, ∞]  0∞ := 0

τ is the map that has next condition.

(i) τ ( A+B ) =τA +τB,   ∀A,B∈N

(ii) τ (λA ) = λτ ( A )      ∀A∈N_+,   ∀λ∈[0, ∞)

(iii) τ ( A*A ) = τ ( AA* )   ∀A∈N

<1-2>

Trace over von Neumann algebra N          τ

(1) τ is faithful.     A∈N, τ (A) = 0 → A = 0

(2) τ is normal.     Increase net {A_n} ⊂N₊   τ (sup_α A_α) = sup_α τ (Aα)

(3) τ is definite.    τ (I ) < ∞

(4) τ is semi-definite.     When A(≠0)∈N_+,    there exist B(≠0) ∈N₊  while B≦A and τ (B) ≠0.

To be continued

Tokyo May 1, 2008

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