L^p-operator algebras associated with oriented graphs

2017/12/23 by Guillermo Cortiñas, Cortiñas, Guillermo, Ma. Eugenia Rodrí guez +1 · 1 citation
Mathematics · #FOS: Mathematics #Operator Algebras (math.OA) #math.OA

paper · pdf · doi:10.48550/arxiv.1712.08824

21 pages

arxiv created 2018/01/19 · arxiv updated 2018/01/22

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 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 R_n the graph with one vertex and n loops (2≤ n≤ ∞), O^p(R_n)=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_tight^p(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 L_Q is simple, and that in this case it is isometrically isomorphic to the closure ρ(L_Q) of the image of any nonzero spatial L^p-representation ρ:L_Q→\mathscrL(L^p(X)). We also show that if L_Q is purely infinite simple and p≠ p', then there is no nonzero continuous homomorphism O^p(Q)\toO^p'(Q). Our results generalize those obtained by Phillips for L^p-Cuntz algebras.

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