There is no separable universal 𝐼𝐼₁-factor

Ozawa, Narutaka

doi:10.1090/S0002-9939-03-07127-2

There is no separable universal $\mathrm {II}_1$-factor
HTML articles powered by AMS MathViewer

by Narutaka Ozawa PDF

Proc. Amer. Math. Soc. 132 (2004), 487-490 Request permission

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

A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869
Pierre de la Harpe and Alain Valette, La propriété $(T)$ de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger), Astérisque 175 (1989), 158 (French, with English summary). With an appendix by M. Burger. MR 1023471
Eberhard Kirchberg, On nonsemisplit extensions, tensor products and exactness of group $C^*$-algebras, Invent. Math. 112 (1993), no. 3, 449–489. MR 1218321, DOI 10.1007/BF01232444
Eberhard Kirchberg, On restricted perturbations in inverse images and a description of normalizer algebras in $C^*$-algebras, J. Funct. Anal. 129 (1995), no. 1, 1–34. MR 1322640, DOI 10.1006/jfan.1995.1040
A. Yu. Ol′shanskiĭ, On residualing homomorphisms and $G$-subgroups of hyperbolic groups, Internat. J. Algebra Comput. 3 (1993), no. 4, 365–409. MR 1250244, DOI 10.1142/S0218196793000251
S. Popa, Correspondences, Preprint 1986.
Sorin Popa, Some rigidity results in type $\textrm {II}_1$ factors, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 9, 535–538 (English, with French summary). MR 1078117
Shôichirô Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York-Heidelberg, 1971. MR 0442701
Alain Valette, Old and new about Kazhdan’s property (T), Representations of Lie groups and quantum groups (Trento, 1993) Pitman Res. Notes Math. Ser., vol. 311, Longman Sci. Tech., Harlow, 1994, pp. 271–333. MR 1431309