Free products of von Neumann algebras
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- by Wai Mee Ching PDF
- Trans. Amer. Math. Soc. 178 (1973), 147-163 Request permission
Abstract:
A new method of constructing factors of type ${\text {II}_1}$, called free product, is introduced. It is a generalization of the group construction of factors of type ${\text {II}_1}$ when the given group is a free product of two groups. If ${A_1}$ and ${A_2}$ are two von Neumann algebras with separating cyclic trace vectors and ortho-unitary bases, then the free product ${A_1} \ast {A_2}$ of ${A_1}$ and ${A_2}$ is a factor of type ${\text {II}_1}$ without property $\Gamma$.References
-
H. Behncke, Automorphisms of $\mathcal {A}({\Phi _2})$, notes.
- Robert J. Blattner, Automorphic group representations, Pacific J. Math. 8 (1958), 665–677. MR 103421, DOI 10.2140/pjm.1958.8.665
- Wai-mee Ching, Non-isomorphic non-hyperfinite factors, Canadian J. Math. 21 (1969), 1293–1308. MR 254614, DOI 10.4153/CJM-1969-142-6
- Wai-mee Ching, A continuum of non-isomorphic non-hyperfinite factors, Comm. Pure Appl. Math. 23 (1970), 921–937. MR 279593, DOI 10.1002/cpa.3160230605
- Jacques Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien (Algèbres de von Neumann), Cahiers Scientifiques, Fasc. XXV, Gauthier-Villars, Paris, 1957 (French). MR 0094722
- J. Dixmier and E. C. Lance, Deux nouveaux facteurs de type $\textrm {II}_{1}$, Invent. Math. 7 (1969), 226–234 (French). MR 248535, DOI 10.1007/BF01404307
- Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
- Dusa McDuff, A countable infinity of $\Pi _{1}$ factors, Ann. of Math. (2) 90 (1969), 361–371. MR 256183, DOI 10.2307/1970729
- Dusa McDuff, Uncountably many $\textrm {II}_{1}$ factors, Ann. of Math. (2) 90 (1969), 372–377. MR 259625, DOI 10.2307/1970730
- F. J. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), no. 1, 116–229. MR 1503275, DOI 10.2307/1968693
- F. J. Murray and J. von Neumann, On rings of operators. IV, Ann. of Math. (2) 44 (1943), 716–808. MR 9096, DOI 10.2307/1969107
- Masahiro Nakamura and Zirô Takeda, On some elementary properties of the crossed products of von Neumann algebras, Proc. Japan Acad. 34 (1958), 489–494. MR 107828
- L. Pukánszky, Some examples of factors, Publ. Math. Debrecen 4 (1956), 135–156. MR 80894
- Shôichirô Sakai, Asymptotically abelian $\textrm {II}_{1}$-factors, Publ. Res. Inst. Math. Sci. Ser. A 4 (1968/1969), 299–307. MR 0248533, DOI 10.2977/prims/1195194878
- Shôichirô Sakai, An uncountable number of $\textrm {II}_{1}$ and $\textrm {II}_{\infty }$ factors, J. Functional Analysis 5 (1970), 236–246. MR 0259626, DOI 10.1016/0022-1236(70)90028-5
- J. Schwartz, Two finite, non-hyperfinite, non-isomorphic factors, Comm. Pure Appl. Math. 16 (1963), 19–26. MR 149322, DOI 10.1002/cpa.3160160104
- Noboru Suzuki, Crossed products of rings of operators, Tohoku Math. J. (2) 11 (1959), 113–124. MR 105624, DOI 10.2748/tmj/1178244632
- M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics, Vol. 128, Springer-Verlag, Berlin-New York, 1970. MR 0270168, DOI 10.1007/BFb0065832
- Takasi Turumaru, Crossed product of operator algebra, Tohoku Math. J. (2) 10 (1958), 355–365. MR 102764, DOI 10.2748/tmj/1178244669
- J. von Neumann, On infinite direct products, Compositio Math. 6 (1939), 1–77. MR 1557013
- G. Zeller-Meier, Deux nouveaux facteurs de type $\textrm {II}_{1}$, Invent. Math. 7 (1969), 235–242 (French). MR 248536, DOI 10.1007/BF01404308
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 178 (1973), 147-163
- MSC: Primary 46L10
- DOI: https://doi.org/10.1090/S0002-9947-1973-0326405-3
- MathSciNet review: 0326405