Powers' property
Lλ is strengthened by requiring the simultaneous validity over a finite number of states. It is then shown that a von Neumann algebra
R on a separable space has the modified property—called the property
Lλ′—if and only if λ(1−λ)
-1 is in the asymptotic set r
∞(
R), where 0{≤}λ{≤}1/2. It is also noted that any finite continuous von Neumann algebra has the property
L1/2.
The closedness of r
∞(
R) for any von Neumann algebra
R on a separable space follows as a corollary.
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