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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
DOI: https://doi.org/10.1090/S0002-9947-1973-0317062-0
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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DOI: https://doi.org/10.1090/S0002-9947-1973-0317062-0
Keywords: Asymptotic abelianness, type III factor, type $ {\text{II}_\infty }$ factor, property L, central sequence, equivalent sequences, trivial sequence, $ ^\ast$-automorphism, invariant state, strong convergence, uniform convergence
Article copyright: © Copyright 1973 American Mathematical Society