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.
Categories: math.OA