Rigidity of ${\ell }^p$ Roe-type algebras

Abstract

We investigate the rigidity of the ${\ell }^p$ analog of Roe-type algebras. In particular, we show that if $p\in [1,\infty )\setminus \{2\}$, then an isometric isomorphism between the ${\ell }^p$ uniform Roe algebras of two metric spaces with bounded geometry yields a bijective coarse equivalence between the underlying metric spaces, while a stable isometric isomorphism yields a coarse equivalence. We also obtain similar results for other ${\ell }^p$ Roe-type algebras. In this paper, we do not assume that the metric spaces have Yu's property A or finite decomposition complexity.

Final version, 19 pages. To appear in Bulletin of the London Mathematical Society. Corrected minor typos, and updated bibliography