JournalsjncgVol. 15, No. 2pp. 581–614

Structure and KK-theory of p\ell^p uniform Roe algebras

  • Yeong Chyuan Chung

    Polish Academy of Sciences, Warsaw, Poland
  • Kang Li

    Polish Academy of Sciences, Warsaw, Poland
Structure and $K$-theory of $\ell^p$ uniform Roe algebras cover
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Abstract

In this paper, we characterize when the p\ell^p uniform Roe algebra of a metric space with bounded geometry is (stably) finite and when it is properly infinite in standard form for p[1,)p\in [1,\infty). Moreover, we show that the p\ell^p uniform Roe algebra is a (non-sequential) spatial LpL^p AF algebra in the sense of Phillips and Viola if and only if the underlying metric space has asymptotic dimension zero.

We also consider the ordered K0K_0 groups of p\ell^p uniform Roe algebras for metric spaces with low asymptotic dimension, showing that (1) the ordered K0K_0 group is trivial when the metric space is non-amenable and has asymptotic dimension at most one, and (2) when the metric space is a countable locally finite group, the (ordered) K0K_0 group is a complete invariant for the (bijective) coarse equivalence class of the underlying locally finite group. It happens that in both cases the ordered K0K_0 group does not depend on p[1,)p\in [1,\infty).

Cite this article

Yeong Chyuan Chung, Kang Li, Structure and KK-theory of p\ell^p uniform Roe algebras. J. Noncommut. Geom. 15 (2021), no. 2, pp. 581–614

DOI 10.4171/JNCG/405