  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.
