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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
Published electronically: June 23, 2003
MathSciNet review: 2022373
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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.

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Additional Information

Narutaka Ozawa
Affiliation: Department of Mathematical Science, University of Tokyo, Tokyo 153-8914, Japan

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

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