Simplicity of $L^p$-graph algebras

TL;DR

This paper establishes an equivalence between the simplicity of Leavitt path algebras and their associated $L^p$-operator graph algebras, revealing a fundamental connection in their algebraic structures.

Contribution

It demonstrates that the simplicity of $L^p$-operator graph algebras is equivalent to the simplicity of Leavitt path algebras for all $p$, unifying their structural understanding.

Findings

Simplicity of $ ext{O}^p(E)$ is equivalent to that of $L(E)$.

Purely infinite simplicity is characterized similarly in both algebra types.

Results hold for all $p$ in $[1, \infty)$.

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.