Asymptotic Ratio Set and Property $L'_λ$

Abstract

Powers' property Lx 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 L'r-if and only if A(l—x)- 1 is in the asymptotic set r^CR). where O^A^l/2. It is also noted that any finite continuous von Neumann algebra has the property Li/z. The closedness of r «,(./?) for any von Neumann algebra R on a separable space follows as a corollary.