Wednesday, 17 July 2013

Purely Infinite


Note 2
Purely Infinite  



[Theorem]
The necessary and sufficient condition for what von Neumann algebra N is purely infinite ( type) 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,BN
(ii) τ (λA ) = λτ A )      AN+,   λ[0, ∞)
(iii) τ A*A ) = τ AA* )   AN
<1-2>
Trace over von Neumann algebra N          τ
(1) τ is faithful.     ANτ (A) = 0  A = 0
(2) τ is normal.     Increase net {AnN+   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 BA and τ (B0.

To be continued
Tokyo May 1, 2008

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