Free products of von Neumann algebras

Ching, Wai Mee

doi:10.1090/S0002-9947-1973-0326405-3

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

Automorphisms of

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