×

Duality for crossed products and the structure of von Neumann algebras of type III. (English) Zbl 0268.46058


MSC:

46L10 General theory of von Neumann algebras
Full Text: DOI

References:

[1] Araki, H., A classifiction of factors, II.Publications of Research Institute for Math. Sciences, Kyoto University, Ser. A, 4 (1968), 585–593. · Zbl 0198.17803 · doi:10.2977/prims/1195194819
[2] Araki, H.,Structure of some von Neumann algebras with isolated discrete modular spectrum. To appear. · Zbl 0269.46048
[3] Araki, H. &Woods, J., A classification of factors.Publications of Research Institute for Math. Sciences, Kyoto University, Ser. A, 4 (1968), 51–130. · Zbl 0206.12901 · doi:10.2977/prims/1195195263
[4] Arveson, W.,On groups of automorphisms of operator algebras. To appear. · Zbl 0296.46064
[5] Blattner, R. J., On induced representations.Amer. J. Math., 83 (1961), 79–98. · Zbl 0122.28405 · doi:10.2307/2372722
[6] –, On a theorem of G. W. Mackey.Bull. Amer. Math. Soc., 68 (1962), 585–587. · Zbl 0122.35202 · doi:10.1090/S0002-9904-1962-10859-3
[7] Bourbaki, N.,Intégration, Chap. 1–4, Paris (1952). · Zbl 0049.31703
[8] Bourbaki, N.,Intégration, Chap. 7–8, Paris (1963). · Zbl 0156.03204
[9] Combes, F., Poids associés à une algèbre hilbertienne à gauche.Comp. Math., 23 (1971), 49–77. · Zbl 0208.38201
[10] Connes, A.,Une classification des facteurs de type III. Thesis, to appear. · Zbl 0243.46064
[11] Dixmier, J.,Les algèbres d’opérateurs dans l’espace hilbertien, 2nd edition, Paris, Gauthier-Villars, (1969). · Zbl 0175.43801
[12] Dotlicher, S., Kaspler, D. &Robinson, D., Covariance algebras in field theory and statistical mechanics.Comm. Math. Phys., 3 (1966), 1–28. · Zbl 0152.23803 · doi:10.1007/BF01645459
[13] Haga, Y.,On subalgebras of a crossed product von Neumann algebra. To appear. · Zbl 0268.46059
[14] Haga, Y. & Takeda, Z., Correspondence between subgroups and subalgebras in a cross product von Neumann algebra. To appear inTôhoku Math. J. · Zbl 0246.46052
[15] Herman, R. &Takesaki, M., States and automorphism groups of operator algebras.Comm. Math. Phys., 19 (1940), 142–160. · Zbl 0206.13001 · doi:10.1007/BF01646631
[16] Ionescu Tulcea, A. andIonescu Tulcea, C.,Topics in the theory of lifting. Springer-Verlag, New York, (1969). · Zbl 0179.46303
[17] Ionescu Tulcea, A. andIonescu Tulcea, C.,On the existence of a lifting commuting with the left translations of an arbitrary locally compact groups, Proc. Fifth Berkeley Symp. in Math. Stat. and Prob. Univ. of California Press, (1967), 63–97. · Zbl 0201.49202
[18] Loomis, L. H., Note on a theorem of Mackay.Duke Math. J., 19 (1952), 641–645. · Zbl 0047.35501 · doi:10.1215/S0012-7094-52-01968-6
[19] Mackey, G. W., A theorem of Stone and von Neumann.Duke Math. J., 16 (1949), 313–326. · Zbl 0036.07703 · doi:10.1215/S0012-7094-49-01631-2
[20] –, Induced representations of locally compact groups, I.Ann. Math., 55 (1952), 101–139. · Zbl 0046.11601 · doi:10.2307/1969423
[21] –, Unitary representations of group extensions.Acta Math., 99 (1958), 265–311. · Zbl 0082.11301 · doi:10.1007/BF02392428
[22] Maréchal, O., Champs measureables d’espaces hilbertiens.Bull. Sc. Math., 93 (1969), 113–143. · Zbl 0186.45601
[23] McDuff, D., A countable infinity of II1 factors.Ann. Math., 90 (1969), 361–371. · Zbl 0184.16901 · doi:10.2307/1970729
[24] –, Uncountably many II1 factors.Ann. Math., 90 (1969), 372–377. · Zbl 0184.16902 · doi:10.2307/1970730
[25] Murray, F. J. &von Neumann, J., On rings of operators.Ann. Math., 37 (1936), 116–229. · Zbl 0014.16101 · doi:10.2307/1968693
[26] –, On rings of operators IV.Ann. Math., 44 (1943), 716–808. · Zbl 0060.26903 · doi:10.2307/1969107
[27] Nakamura, M. &Takeda, Z., On some elementary properties of the crossed products of von Neumann algebras.Proc. Japan Acad., 34 (1958), 489–494. · Zbl 0085.09905 · doi:10.3792/pja/1195524559
[28] –, A Galois theory for finite factors.Proc. Japan Acad., 36 (1960), 258–260. · Zbl 0098.30804 · doi:10.3792/pja/1195524026
[29] –, On the fundamental theorem of the Galois theory for finite factors.Proc. Japan Acad., 36 (1960), 313–318. · Zbl 0094.09803 · doi:10.3792/pja/1195523999
[30] Nakamura, M. &Umegaki, H., Heisenberg’s commutation relation and the Plancherel theorem.Proc. Japan Acad., 37 (1961), 239–242. · Zbl 0101.09404 · doi:10.3792/pja/1195523709
[31] von Neumann, J., Die Eindeutigkeit der Schrödingerschen Operatoren.Math. Ann., 104 (1931), 570–578. · Zbl 0001.24703 · doi:10.1007/BF01457956
[32] Nielsen, O. A.,The Mackey-Blattner theorem and Takesaki’s generalized commutation relation for locally compact groups. To appear. · Zbl 0256.22010
[33] Pedersen, G. K. &Takesaki, M., The Radon-Nikodym theorem for von Neumann algebras.Acta Math., 130 (1973), 53–87. · Zbl 0262.46063 · doi:10.1007/BF02392262
[34] Powers, R., Representations of uniformly hyperfinite algebras and their associated von Neumann rings.Ann. Math., 86 (1967), 138–171. · Zbl 0157.20605 · doi:10.2307/1970364
[35] Rudin, W.,Fourier analysis on groups. Intersciences, New York, (1962). · Zbl 0107.09603
[36] Sakai, S., An uncountable number of II1 and IIfactors.J. Functional Analysis, 5 (1970), 236–246. · Zbl 0198.17804 · doi:10.1016/0022-1236(70)90028-5
[37] Størmer, E.,Spectra of states, and asymptotically abelian C *-algebras. To appear. · Zbl 0245.46085
[38] Suzuki, N., Cross products of rings of operators.Tôhoku Math. J., 11 (1959), 113–124. · Zbl 0106.31005 · doi:10.2748/tmj/1178244632
[39] Takesaki, M., Covariant representations ofC *-algebras and their locally compact automorphism groups.Acta Math., 119 (1967), 273–303. · Zbl 0163.36802 · doi:10.1007/BF02392085
[40] –, A liminal crossed product of a uniformly hyperfiniteC *-algebra by a compact abelian automorphism group.J. Functional Analysis, 7 (1971), 140–146. · Zbl 0229.46055 · doi:10.1016/0022-1236(71)90049-8
[41] –, A generalized commutation relation for the regular representation.Bull. Soc. Math., France, 97 (1969), 289–297. · Zbl 0188.20101
[42] Takesaki, M.,Tomita’s theory of modular Hilbert algebras and its application. Lecture Notes in Mathematics, 128, Springer-Verlag, (1970). · Zbl 0193.42502
[43] Takesaki, M.,States and automorphisms of operator algebras–Standard representations and the Kubo-Martin-Schwinger boundary condition. Summer Rencontres in Mathematics and Physics, Battelle Seattle, (1971). · Zbl 0258.46055
[44] Takesaki, M., Periodic and homogeneous states on a von Neumann algebras, I, II, III. To appear inBull. Amer. Math. Soc.
[45] –, The structure of a von Neumann algebra with a homogeneous periodic state.Acta Math., 131 (1973), 79–121. · Zbl 0267.46047 · doi:10.1007/BF02392037
[46] –, Dualité dans les produits croisés d’algebres de von Neumann.C. R. Acad. Sc. Paris, Ser. A, 276 (1973), 41–43.
[47] Takesaki, M., Algèbres de von Neumann proprement infinits et produits croisés.C. R. Acad. Sc. Paris, Ser. A, 276 (1973). · Zbl 0247.46068
[48] Takesaki, M., Duality in crossed products and von Neumann algebras of type III. To appearBull. Amer. Math. Soc. · Zbl 0269.46047
[49] Takesaki, M. &Tatsuuma, N., Duality and subgroups.Ann. Math., 93 (1971), 344–364. · doi:10.2307/1970778
[50] Tomita, M., Standard forms of von Neumann algebras.The 5th Functional Analysis Symposium of the Math. Soc. of Japan, Sendai, (1967).
[51] Turumaru, T., Crossed product of operator algebras.Tôhoka Math. J., 10 (1958), 355–365. · Zbl 0087.31803 · doi:10.2748/tmj/1178244669
[52] Vesterstrøm, J. &Wils, W., Direct integrals of Hilbert spaces, II.Math. Scand., 26 (1970), 89–102. · Zbl 0207.44005
[53] Zeller-Meier, G., Produits croisés d’uneC *-algebre par un groupe d’automorphismes.J. Math. Pures et appl., 47 (1968), 101–239. · Zbl 0165.48403
[54] A. Connes & Van Daele, The group property of the invariant S. To appear inMath. Scand. · Zbl 0268.46056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.