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Simple Lie groups without the Approximation Property II


Authors: Uffe Haagerup and Tim de Laat
Journal: Trans. Amer. Math. Soc. 368 (2016), 3777-3809
MSC (2010): Primary 22D25, 46B28; Secondary 46L07
DOI: https://doi.org/10.1090/tran/6448
Published electronically: July 14, 2015
MathSciNet review: 3453357
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Abstract: We prove that the universal covering group $ \widetilde {\mathrm {Sp}}(2,\mathbb{R})$ of $ \mathrm {Sp}(2,\mathbb{R})$ does not have the Approximation Property (AP). Together with the fact that $ \mathrm {SL}(3,\mathbb{R})$ does not have the AP, which was proved by Lafforgue and de la Salle, and the fact that $ \mathrm {Sp}(2,\mathbb{R})$ does not have the AP, which was proved by the authors of this article, this finishes the description of the AP for connected simple Lie groups. Indeed, it follows that a connected simple Lie group has the AP if and only if its real rank is zero or one. By an adaptation of the methods we use to study the AP, we obtain results on approximation properties for noncommutative $ L^p$-spaces associated with lattices in $ \widetilde {\mathrm {Sp}}(2,\mathbb{R})$. Combining this with earlier results of Lafforgue and de la Salle and results of the second-named author of this article, this gives rise to results on approximation properties of noncommutative $ L^p$-spaces associated with lattices in any connected simple Lie group.


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Additional Information

Uffe Haagerup
Affiliation: Department of Mathematical Sciences, University of Copenhagen Universitetsparken 5, DK-2100 Copenhagen O, Denmark

Tim de Laat
Affiliation: KU Leuven, Department of Mathematics, Celestijnenlaan 200B – Box 2400, B-3001 Leuven, Belgium
Email: tim.delaat@wis.kuleuven.be

DOI: https://doi.org/10.1090/tran/6448
Received by editor(s): July 17, 2013
Received by editor(s) in revised form: March 5, 2014
Published electronically: July 14, 2015
Additional Notes: The first-named author was supported by ERC Advanced Grant no. OAFPG 247321, the Danish Natural Science Research Council, and the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).
The second-named author was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).
Sadly, Uffe Haagerup passed away on July 5, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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