$L^p$-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 ($2\le n\le \infty$), ${\mathcal{O}}^p({\mathcal{R}}_n)={\mathcal{O}}_n^p$, the $L^p$-Cuntz algebra introduced by Phillips. If $p\notin \{1,2\}$ and $\mathcal{S}(Q)$ is the inverse semigroup generated by $Q$, ${\mathcal{O}}^p(Q)=F_{\mathrm{tight}}^p(\mathcal{S}(Q))$ 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 $\rho :L_Q\to \mathcal{L}(L^p(X))$. We also show that if $L_Q$ is purely infinite simple and $p\ne p^{\prime }$, then there is no nonzero continuous homomorphism ${\mathcal{O}}^p(Q)\to {\mathcal{O}}^{p^{\prime }}(Q)$. Our results generalize those obtained by Phillips for $L^p$-Cuntz algebras.

21 pages

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