  In this paper, we characterize when the $\ell^p$ uniform Roe algebra of a
metric space with bounded geometry is (stably) finite and when it is properly
infinite in standard form for $p\in [1,\infty)$. Moreover, we show that the
$\ell^p$ uniform Roe algebra is a (non-sequential) spatial $L^p$ 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 $K_0$ groups of $\ell^p$ uniform Roe algebras
for metric spaces with low asymptotic dimension, showing that (1) the ordered
$K_0$ 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) $K_0$ 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 $K_0$ group does not depend on $p\in
[1,\infty)$.
