$L^p$-operator algebras associated with oriented graphs

Abstract

For each $1\le p<\infty$ and each countable oriented graph $Q$ we introduce an $L^p$-operator algebra $\mathcal{O}^p(Q)$ which contains the Leavitt path $\mathbb{C}$-algebra $L_Q$ as a dense subalgebra and is universal for those $L^p$-representations of $L_Q$ which are spatial in the sense of N.C. Phillips. For $\mathcal{R}_n$ the graph with one vertex and $n$ loops ($2\le n\le \infty$), $\mathcal{O}^p(\mathcal{R}_n)=\mathcal{O}^p_n$, the $L^p$-Cuntz algebra introduced by Phillips. If $p\notin\{1,2\}$ and $\mathcal{S}(Q)$ is the inverse semigroup generated by $Q$, $\mathcal{O}^p(Q)=F_{\operatorname{tight}}^p(\mathcal{S}(Q))$ is the tight semigroup $L^p$-operator algebra introduced by Gardella and Lupini. We prove that $\mathcal{O}^p(Q)$ is simple as an $L^p$-operator algebra if and only if $L_Q$ is simple, and that in this case it is isometrically isomorphic to the closure $\overline{\rho(L_Q)}$ of the image of any nonzero spatial $L^p$-representation $\rho:L_Q\to\mathscr{L}(L^p(X))$. We also show that if $L_Q$ is purely infinite simple and $p\ne p'$, then there is no nonzero continuous homomorphism $\mathcal{O}^p(Q)\to\mathcal{O}^{p'}(Q)$. Our results generalize those obtained by Phillips for $L^p$-Cuntz algebras.