Ultraproducts of von Neumann algebras
Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
Standard
Ultraproducts of von Neumann algebras. / Ando, Hiroshi; Haagerup, Uffe.
I: Journal of Functional Analysis, Bind 266, Nr. 12, 2014, s. 6842-6913.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
Harvard
APA
Vancouver
Author
Bibtex
}
RIS
TY - JOUR
T1 - Ultraproducts of von Neumann algebras
AU - Ando, Hiroshi
AU - Haagerup, Uffe
PY - 2014
Y1 - 2014
N2 - 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 MωMω introduced by Ocneanu is a corner of the ultraproduct ∏ωM∏ωM 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 (σtφω=(σtφ)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 MωMω 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 MωMω and Connes' asymptotic centralizer algebra MωMω.
AB - 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 MωMω introduced by Ocneanu is a corner of the ultraproduct ∏ωM∏ωM 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 (σtφω=(σtφ)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 MωMω 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 MωMω and Connes' asymptotic centralizer algebra MωMω.
U2 - 10.1016/j.jfa.2014.03.013
DO - 10.1016/j.jfa.2014.03.013
M3 - Journal article
VL - 266
SP - 6842
EP - 6913
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 12
ER -
ID: 137751676