Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Property $ L$ and asymptotic abelianness for $ W^*$-algebras


Author: Edward Sarian
Journal: Proc. Amer. Math. Soc. 80 (1980), 125-129
MSC: Primary 46L10
MathSciNet review: 574521
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: (i) Let $ \mathcal{A}$ be a finite $ W^*$-algebra acting on a separable Hilbert space and having no abelian direct summand. If $ \mathcal{A}$ is asymptotically abelian, then $ \mathcal{A}$ has property L. (ii) Let $ \mathcal{A}$ be a finite $ W^*$-algebra acting on a separable Hilbert space. Then $ \mathcal{A} \otimes B(h), h$ a separable infinite dimensional Hilbert space, is not asymptotically abelian. (iii) Type $ \mathrm{II}_\infty W^*$-algebras are not asymptotically abelian. (iv) Noncommutative type I $ W^*$-algebras are not asymptotically abelian. (v) The type III factor $ \mathcal{B} = \mathcal{P} \otimes \mathcal{A}(G)$ is not asymptotically abelian. $ \mathcal{B}$ produces uncountably many nonisomorphic nonasymptotically abelian factors of type III and establishes an example of a purely infinite factor that has property L but is not asymptotically abelian.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46L10

Retrieve articles in all journals with MSC: 46L10


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0574521-2
PII: S 0002-9939(1980)0574521-2
Keywords: Property L, asymptotic abeliannness, finite $ W^*$-algebra, type I $ W^*$-algebra, type $ \mathrm{II}_\infty W^*$-algebra, type III factor, central trace, central sequence, $ *$-automorphism
Article copyright: © Copyright 1980 American Mathematical Society