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Embeddings of Uniform Roe Algebras

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Abstract

In this paper, we study embeddings of uniform Roe algebras. Generally speaking, given metric spaces X and Y, we are interested in which large scale geometric properties are stable under embedding of the uniform Roe algebra of X into the uniform Roe algebra of Y.

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Notes

  1. Recall, a coarse map \(f:(X,d)\rightarrow (Y,\partial )\) is a map so that for all \(s>0\) there exists \(r>0\) such that \(d(x,y)<s\) implies \(\partial (f(x),f(y))<r\). See Sect. 2.

  2. This property for metric spaces is often called bounded geometry in the literature.

  3. In [24, Lemma 5.2] X is assumed to have the metric sparsification property, but this is equivalent to property A for u.l.f. metric spaces (see [6, Propositioshn 4.1], [5, Proposition 3.2 and Theorem 3.8] and [19, Theorem 4.1]).

  4. More precisely, in [4, Lemma 3.4] one has \(\mathrm {C}^*_u(Y)\) instead of \(\mathcal {B}(\ell _2(Y))\). However, this is not used in the proof.

  5. An ideal \(\mathscr {I}\subseteq \mathcal P(X)\) is ccc/Fin if every family of almost disjoint sets which are not in \(\mathscr {I}\) is countable.

  6. We refer the reader to [4] for a detailed study of uniform Roe coronas.

  7. Precisely, the proof of [4, Proposition 4.1] starts by taking a an infinite \(L\subseteq \mathbb {N}\) and working with the subsequence \((p_n)_{n\in L}\). Since \(\mathscr {I}\) is ccc/Fin, going to a further subsequence we can suppose that \(L\in \mathscr {I}\). The proof now holds verbatim.

  8. Notice that it is not known whether FDC and property A are equivalent.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada

    Bruno M. Braga & Ilijas Farah

  2. Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001, Leuven, Belgium

    Alessandro Vignati

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  1. Bruno M. Braga
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  2. Ilijas Farah
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  3. Alessandro Vignati
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Correspondence to Bruno M. Braga.

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Communicated by Y. Kawahigashi

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Braga, B.M., Farah, I. & Vignati, A. Embeddings of Uniform Roe Algebras. Commun. Math. Phys. 377, 1853–1882 (2020). https://doi.org/10.1007/s00220-019-03539-9

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