Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

There is no separable universal $\mathrm{II}_1$-factor


Author: Narutaka Ozawa
Journal: Proc. Amer. Math. Soc. 132 (2004), 487-490
MSC (2000): Primary 46L10; Secondary 20F65
DOI: https://doi.org/10.1090/S0002-9939-03-07127-2
Published electronically: June 23, 2003
MathSciNet review: 2022373
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Gromov constructed uncountably many pairwise nonisomorphic discrete groups with Kazhdan's property $\mathrm{(T)}$. We will show that no separable $\mathrm{II}_1$-factor can contain all these groups in its unitary group. In particular, no separable $\mathrm{II}_1$-factor can contain all separable $\mathrm{II}_1$-factors in it. We also show that the full group $C^*$-algebras of some of these groups fail the lifting property.


References [Enhancements On Off] (What's this?)

  • [C] A. Connes, Classification of injective factors. Cases $II_1$, $II_\infty$, $III_\lambda$, $\lambda\neq1$, Ann. of Math. (2) 104 (1976), no. 1, 73-115. MR 56:12908
  • [G] M. Gromov, Hyperbolic groups, Essays in group theory, 75-263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987. MR 89e:20070
  • [H] P. de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 2001i:20081
  • [HV] P. de la Harpe and A. Valette, La propriété $(T)$ de Kazhdan pour les groupes localement compacts, With an appendix by M. Burger. Astérisque 175 (1989), 158 pp. MR 90m:22001
  • [K1] E. Kirchberg, On nonsemisplit extensions, tensor products and exactness of group $C^*$-algebras, Invent. Math. 112 (1993), no. 3, 449-489. MR 94d:46058
  • [K2] E. Kirchberg, On subalgebras of the CAR-algebra, J. Funct. Anal. 129 (1995), no. 1, 35-63. MR 95m:46094b
  • [O] A. Yu. Olshanskii, On residualing homomorphisms and $G$-subgroups of hyperbolic groups. Internat. J. Algebra Comput. 3 (1993), no. 4, 365-409. MR 94i:20069
  • [P1] S. Popa, Correspondences, Preprint 1986.
  • [P2] S. Popa, Some rigidity results in type $\mathrm{II}_1$ factors, C. R. Acad. Sci. Paris Ser. I Math. 311 (1990), 535-538. MR 92e:46123
  • [S] S. Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60. Springer-Verlag, New York-Heidelberg, 1971. MR 56:1082
  • [V] A. Valette, Old and new about Kazhdan's property $\mathrm{(T)}$, Representations of Lie groups and quantum groups (Trento, 1993), 271-333, Pitman Res. Notes Math. Ser., 311, Longman Sci. Tech., Harlow, 1994. MR 98a:22003

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46L10, 20F65

Retrieve articles in all journals with MSC (2000): 46L10, 20F65


Additional Information

Narutaka Ozawa
Affiliation: Department of Mathematical Science, University of Tokyo, Tokyo 153-8914, Japan
Email: narutaka@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-03-07127-2
Keywords: Universal $\mathrm{II}_1$-factor, uncountably many $\mathrm{II}_1$-factors, lifting property
Received by editor(s): October 10, 2002
Published electronically: June 23, 2003
Additional Notes: The author was partially supported by JSPS Postdoctoral Fellowships for Research Abroad.
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society