  We prove that, for $p \in [1, \infty),$ and integers $d$ at least 2, the
$L^p$ analog ${\mathcal{O}}_d^p$ of the Cuntz algebra ${\mathcal{O}}_d$ is a
purely infinite simple amenable Banach algebra.
  The proof requires what we call the spatial $L^p$ UHF algebras, which are
analogs of UHF algebras acting on $L^p$ spaces. As for the usual UHF
C*-algebras, they have associated supernatural numbers. For fixed $p \in [1,
\infty),$ we prove that any spatial $L^p$ UHF algebra is simple and amenable,
and that two such algebras are isomorphic if and only if they have the same
supernatural number (equivalently, the same scaled ordered $K_0$-group). For
distinct $p_1, p_2 \in [1, \infty),$ we prove that no spatial $L^{p_1}$ UHF
algebra is isomorphic to any spatial $L^{p_2}$ UHF algebra.
