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- Title
Approximation properties for noncommutative L<sup>p</sup>-spaces of high rank lattices and nonembeddability of expanders.
- Authors
de Laat, Tim; de la Salle, Mikael
- Abstract
This article contains two rigidity type results for SL(n,Z) for large n that share the same proof. Firstly, we prove that for every p ∈ [1,∞] different from 2, the noncommutative Lp-space associated with SL(n,Z) does not have the completely bounded approximation property for sufficiently large n depending on p. The second result concerns the coarse embeddability of expander families constructed from SL(n,Z). Let X be a Banach space and suppose that there exist β < 1/2 and C > 0 such that the Banach-Mazur distance to a Hilbert space of all k-dimensional subspaces of X is bounded above by Ckβ. Then the expander family constructed from SL(n,Z) does not coarsely embed into X for sufficiently large n depending on X. More generally, we prove that both results hold for lattices in connected simple real Lie groups with sufficiently high real rank.
- Subjects
NONCOMMUTATIVE function spaces; BANACH spaces; HILBERT space; LIE groups; LATTICE theory
- Publication
Journal für die Reine und Angewandte Mathematik, 2018, Vol 2018, Issue 737, p49
- ISSN
0075-4102
- Publication type
Article
- DOI
10.1515/crelle-2015-0043