Lᵖ-operator algebras associated with oriented graphs

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For each 1≤ p<∞ and each countable oriented graph Q we introduce an Lᵖ-operator algebra Oᵖ(Q) which contains the Leavitt path C-algebra LQ as a dense subalgebra and is universal for those Lᵖ-representations of LQ which are spatial in the sense of N.C. Phillips. For Rₙ the graph with one vertex and n loops (2≤ n≤ ∞), Oᵖ(Rₙ)=Oᵖₙ, the Lᵖ-Cuntz algebra introduced by Phillips. If p∉{1,2} and S(Q) is the inverse semigroup generated by Q, Oᵖ(Q)=Ftightᵖ(S(Q)) is the tight semigroup Lᵖ-operator algebra introduced by Gardella and Lupini. We prove that Oᵖ(Q) is simple as an Lᵖ-operator algebra if and only if LQ is simple, and that in this case it is isometrically isomorphic to the closure ρ(LQ) of the image of any nonzero spatial Lᵖ-representation ρ:LQ→L(Lᵖ(X)). We also show that if LQ is purely infinite simple and p≠ p', then there is no nonzero continuous homomorphism Oᵖ(Q)ᵖ'(Q). Our results generalize those obtained by Phillips for Lᵖ-Cuntz algebras.