The Effros-Maréchal Topology in the Space of Von Neumann Algebras

New concepts of limes inferior and limes superior in the space of von Neumann algebras on a fixed Hilbert space are defined, and the topology corresponding to these notions is related to earlier works of Effros and Marechal. A main technical result is that the commutant operation is a homeomorphism...

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Bibliographic Details
Published in:American journal of mathematics 1998-06, Vol.120 (3), p.567-617
Main Authors: Haagerup, Uffe, Winsløw, Carl
Format: Article
Language:English
Subjects:
Algebra
Algebraic topology
Hilbert spaces
Homeomorphism
Mathematical theorems
Mathematics
Topological spaces
Topological theorems
Topology
Von Neumann algebra
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Summary:New concepts of limes inferior and limes superior in the space of von Neumann algebras on a fixed Hilbert space are defined, and the topology corresponding to these notions is related to earlier works of Effros and Marechal. A main technical result is that the commutant operation is a homeomorphism on the space of von Neumann algebras with this topology. Further, the topological properties of several classes and types of von Neumann factors (regarded as subspaces) are determined, and also continuity-type results for Tomita-Takesaki theory are proved. Some applications to subfactor theory are given.
ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.1998.0022