There is no equivariant coarse embedding of $L_{p}$ into $\ell_p$

Krzysztof Święcicki

There is no equivariant coarse embedding of $L_{p}$ into $\ell_p$

Metric Geometry 2020-12-02 v1 Functional Analysis Group Theory

Authors: Krzysztof Święcicki

Abstract

In this paper we prove that $L_{p}$ does not admit an equivariant coarse embedding into $\ell_p$ i.e there is no proper, affine, isometric action of $L_{p}$ , viewed as a group under addition with the standard metric $|| . ||_p$ , on $\ell_p$ . This is done by showing that representations of $L_{p}$ into $Isom(\ell_p)$ has to be trivial, which allows us to reduce the question to bi-Lipschitz setting.

Keywords

equivariant coarse embedding banach spaces lp spaces isometric action bi-lipschitz group representations

Cite

@article{arxiv.2012.00097,
  title  = {There is no equivariant coarse embedding of $L_{p}$ into $\ell_p$},
  author = {Krzysztof Święcicki},
  journal= {arXiv preprint arXiv:2012.00097},
  year   = {2020}
}

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