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Une classification des facteurs de type ${\rm III}$
Open accessAbstract
Le problème principal de la théorie des algèbres de von Neumann est celui de la classification à isomorphisme près des algèbres de von Neumann opérant dans un espace hilbertien séparable.Un théorème de von Neumann permet de se limiter au cas des facteurs (i.e. des algèbres de von Neumann dont le centre est réduit aux scalaires).La théorie de la dimension relative des projecteurs (de Murray et von Neumann) répartit les facteurs en types I, II et III.A tout facteur M de type I est associé un entier n€ [1, oo] et M est isomorphe à Fn = S (^n) où Xn est un espace hilbertien de dimension n.Me DufT a montré (en 1968) l'existence d'une infinité continue de facteurs de type II deux à deux non isomorphes.R. T. Powers a montré (en 1967) l'existence d'une infinité continue (RÀ)>.€]O,I[de facteurs de type III deux à deux non isomorphes.De plus, dans [40], E. J. Woods a prouvé l'impossibilité de construire « effectivement » une bijection entre l'ensemble des classes d'isomorphismes de facteurs de type III et l'ensemble des nombres réels.Cependant, dans [2], H. Araki et E. J. Woods ont défini, par référence aux facteurs R, de Powers, deux invariants algébriques r^ et p, et ont donné une classification des facteurs produits tensoriels infinis de facteurs de type 1$ cette classification a été généralisée par W. Krieger [cf.[23] à [27]) aux facteurs construits à partir de transformations ergodiques.Rappelons qu'un facteur est de type 7^ III si et seulement s'il possède une trace normale fidèle semi-finie.La donnée d'un couple (algèbre de von Neumann, trace normale fidèle semi-finie) est équivalente à la donnée VI.Problèmes.VII.Bibliographie.
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Metadata
- Authors
- Connes, A.
- Venue
- Annales scientifiques de l'École normale supérieure
- Publisher
- Societe Mathematique de France
- Published
- 1973
- Volume
- 6
- Issue
- 2
- Pages
- 133-252
- ISSN
- 0012-9593 (ISSN-L)
- 1873-2151
- OpenAlex
- W2562293972
- Cited by
- 553
- References
- 9
- Citations by year
- 2026: 8 · 2025: 18 · 2024: 23 · 2023: 18 · 2022: 12 · 2021: 10 · 2020: 6 · 2019: 39 · 2018: 11 · 2017: 7 · 2016: 19 · 2015: 10 · 2014: 7 · 2013: 6 · 2012: 22
- FWCI
- 9.20
- Type
- journal article
- Language
- fr
- Open access
- bronze
- doi.org
- Locations
- Annales Scientifiques de l École Normale Supérieure (publishedVersion)
- French digital mathematics library (Numdam) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Physical Sciences › Mathematics › Applied Mathematics › Holomorphic and Operator Theory
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Banach Space Theory
- Keywords
- type III factors, von Neumann algebras, classification of von Neumann algebras, ergodic transformations
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