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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Asymptotic abelianness of infinite factors

Author: M. S. Glaser
Journal: Trans. Amer. Math. Soc. 178 (1973), 41-56
MSC: Primary 46L10
MathSciNet review: 0317062
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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.

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Keywords: Asymptotic abelianness, type III factor, type <!– MATH ${\text {II}_\infty }$ –> <IMG WIDTH="38" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="${\text {II}_\infty }$"> factor, property L, central sequence, equivalent sequences, trivial sequence, <IMG WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$^\ast$">-automorphism, invariant state, strong convergence, uniform convergence
Article copyright: © Copyright 1973 American Mathematical Society