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

TL;DR

This paper proves that for certain $ ext{l}^p$ Roe-type algebras, isometric isomorphisms imply coarse geometric equivalences between the underlying metric spaces, extending rigidity results without assuming property A.

Contribution

It establishes rigidity results for $ ext{l}^p$ Roe-type algebras for $p eq 2$, linking algebraic isomorphisms to coarse geometric equivalences without extra assumptions.

Findings

Isometric isomorphisms imply bijective coarse equivalences for $p eq 2$.

Stable isometric isomorphisms imply coarse equivalences.

Results hold without assuming Yu's property A or finite decomposition complexity.

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.