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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Simple Lie groups without the Approximation Property II
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by Uffe Haagerup and Tim de Laat PDF
Trans. Amer. Math. Soc. 368 (2016), 3777-3809 Request permission

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
  • MR Author ID: 997212
  • Email: tim.delaat@wis.kuleuven.be
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 3777-3809
  • MSC (2010): Primary 22D25, 46B28; Secondary 46L07
  • DOI: https://doi.org/10.1090/tran/6448
  • MathSciNet review: 3453357