Lp-operator algebras associated with oriented graphs

Abstract

For each 1 ≤ p < ∞ and each countable oriented graph Q we introduce an L p -operator algebra O p (Q) which contains the Leavitt path C-algebra LQ as a dense subalgebra and is universal for those L p -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 ≤ ∞), O p (Rn) = O p n , the L p -Cuntz algebra introduced by Phillips. If p < {1, 2} and S(Q) is the inverse semigroup generated by Q, O p (Q) = F p tight(S(Q)) is the tight semigroup L p -operator algebra introduced by Gardella and Lupini. We prove that O p (Q) is simple as an L p -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 L p -representation ρ : LQ → L(L p (X)). We also show that if LQ is purely infinite simple and p , p ′ , then there is no nonzero continuous homomorphism O p (Q) → Op ′ (Q). Our results generalize some similar results obtained by Phillips for L p -Cuntz algebras.