  For each $1\le p<\infty$ and each countable directed graph $E$ we consider
the Leavitt path $\mathbb{C}$-algebra $L(E)$ and the $L^p$-operator graph
algebra $\mathcal{O}^p(E)$. We show that the (purely infinite) simplicity of
$\mathcal{O}^p(E)$ as a Banach algebra is equivalent to the (purely infinite)
simplicity of $L(E)$ as a ring.
