Note 2
Purely Infinite
[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 {An} ⊂N+ τ (supα
Aα) = supα 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