Asymptotic Abelianness of Infinite Factors

Abstract

Studying Pukánszky’s type III factor, ${M_2}$, we show that it does not have the property of asymptotic abelianness and discuss how this property is related to property L. We also prove that there are no asymptotic abelian ${\text {II}_\infty }$ factors. The extension (by ampliation) of central sequences in a finite factor, N, to $M \otimes N$ is shown to be central. Also, we give two examples of the reduction (by equivalence) of a central sequence in $M \otimes N$ to a sequence in N. Finally, applying the definition of asymptotic abelianness of ${C^\ast }$-algebras to ${W^\ast }$-algebras leads to the conclusion that all factors satisfying this property are abelian.