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Journal of Operator Theory

Volume 81, Issue 1, Winter 2019  pp. 225-254.

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

Authors:  Guillermo Cortinas (1), Maria Eugenia Rodriguez (2)
Author institution: (1) Departamento de Matematica-Instituto Santalo, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria (1428) Buenos Aires, Argentina
(2) Departamento de Ciencias Exactas, Ciclo Basico Comun, Universidad de Buenos Aires, Ciudad Universitaria, (1428) Buenos Aires, Argentina

Summary: For each $1\leqslant p<\infty$ 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. We prove that $\mathcal{O}^p(Q)$ is simple as an $L^p$-operator algebra if and only if $L_Q$ is simple, in which case it is isometrically isomorphic to $\overline{\rho(L_Q)}$ for any nonzero spatial $L^p$-representation $\rho: L_Q\to\mathcal{L}(L^p(X))$. If moreover $L_Q$ is purely infinite simple and $p\ne p'$, then there is no nonzero continuous homomorphism $\mathcal{O}^p(Q)\to\mathcal{O}^{p'}(Q)$.

Keywords: oriented graph, Leavitt path algebra, $L^p$-operator algebra, spatial representation, simple, purely infinite, desingularization

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