Ultraproducts of von Neumann algebras
We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Ando, Hiroshi [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2014transfer abstract |
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| Schlagwörter: |
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| Umfang: |
72 |
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| Übergeordnetes Werk: |
Enthalten in: Corrigendum to “Rifampicin resistance mutations in the rpoB gene of - Urusova, Darya V. ELSEVIER, 2022, Amsterdam [u.a.] |
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| Übergeordnetes Werk: |
volume:266 ; year:2014 ; number:12 ; day:15 ; month:06 ; pages:6842-6913 ; extent:72 |
| Links: |
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| DOI / URN: |
10.1016/j.jfa.2014.03.013 |
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| Katalog-ID: |
ELV034260617 |
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| 520 | |a We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . | ||
| 520 | |a We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . | ||
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10.1016/j.jfa.2014.03.013 doi GBVA2014022000023.pica (DE-627)ELV034260617 (ELSEVIER)S0022-1236(14)00133-5 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Ando, Hiroshi verfasserin aut Ultraproducts of von Neumann algebras 2014transfer abstract 72 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . Ultraproducts Elsevier Type III factors Elsevier Haagerup, Uffe oth Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:266 year:2014 number:12 day:15 month:06 pages:6842-6913 extent:72 https://doi.org/10.1016/j.jfa.2014.03.013 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 266 2014 12 15 0615 6842-6913 72 045F 510 |
| spelling |
10.1016/j.jfa.2014.03.013 doi GBVA2014022000023.pica (DE-627)ELV034260617 (ELSEVIER)S0022-1236(14)00133-5 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Ando, Hiroshi verfasserin aut Ultraproducts of von Neumann algebras 2014transfer abstract 72 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . Ultraproducts Elsevier Type III factors Elsevier Haagerup, Uffe oth Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:266 year:2014 number:12 day:15 month:06 pages:6842-6913 extent:72 https://doi.org/10.1016/j.jfa.2014.03.013 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 266 2014 12 15 0615 6842-6913 72 045F 510 |
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10.1016/j.jfa.2014.03.013 doi GBVA2014022000023.pica (DE-627)ELV034260617 (ELSEVIER)S0022-1236(14)00133-5 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Ando, Hiroshi verfasserin aut Ultraproducts of von Neumann algebras 2014transfer abstract 72 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . Ultraproducts Elsevier Type III factors Elsevier Haagerup, Uffe oth Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:266 year:2014 number:12 day:15 month:06 pages:6842-6913 extent:72 https://doi.org/10.1016/j.jfa.2014.03.013 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 266 2014 12 15 0615 6842-6913 72 045F 510 |
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10.1016/j.jfa.2014.03.013 doi GBVA2014022000023.pica (DE-627)ELV034260617 (ELSEVIER)S0022-1236(14)00133-5 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Ando, Hiroshi verfasserin aut Ultraproducts of von Neumann algebras 2014transfer abstract 72 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . Ultraproducts Elsevier Type III factors Elsevier Haagerup, Uffe oth Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:266 year:2014 number:12 day:15 month:06 pages:6842-6913 extent:72 https://doi.org/10.1016/j.jfa.2014.03.013 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 266 2014 12 15 0615 6842-6913 72 045F 510 |
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10.1016/j.jfa.2014.03.013 doi GBVA2014022000023.pica (DE-627)ELV034260617 (ELSEVIER)S0022-1236(14)00133-5 DE-627 ger DE-627 rakwb eng 510 510 DE-600 570 VZ BIODIV DE-30 fid 44.00 bkl Ando, Hiroshi verfasserin aut Ultraproducts of von Neumann algebras 2014transfer abstract 72 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . Ultraproducts Elsevier Type III factors Elsevier Haagerup, Uffe oth Enthalten in Elsevier Urusova, Darya V. ELSEVIER Corrigendum to “Rifampicin resistance mutations in the rpoB gene of 2022 Amsterdam [u.a.] (DE-627)ELV007566018 volume:266 year:2014 number:12 day:15 month:06 pages:6842-6913 extent:72 https://doi.org/10.1016/j.jfa.2014.03.013 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U FID-BIODIV SSG-OLC-PHA 44.00 Medizin: Allgemeines VZ AR 266 2014 12 15 0615 6842-6913 72 045F 510 |
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We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . |
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We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . |
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We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct M ω introduced by Ocneanu is a corner of the ultraproduct ∏ ω M introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state φ of M on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ( σ t φ ω = ( σ t φ ) ω ). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower M ω of a Type III0 factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of M in M ω and Connes' asymptotic centralizer algebra M ω . |
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