CONICET | Buscador de Institutos y Recursos Humanos

For each $1le p<infty$ and each countable oriented graph $Q$ we introduce an $L^p$-operator algebra $cO^p(Q)$, which contains the Leavitt path $C$-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. We prove that $cO^p(Q)$ is simple as an $L^p$-operator algebra if and only if $L_Q$ is simple, in which case it is isometrically isomorphic to $ol{ho(L_Q)}$ for any nonzero spatial $L^p$-representation $hocolon L_QocaL(L^p(X))$. If moreover $L_Q$ is purely infinite simple and $pe p´$, then there is no nonzero continuous homomorphism $cO^p(Q)ocO^{p´}(Q)$.