Simplicity of $L^p$-graph algebras

Abstract

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.