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

MARÍA EUGENIA RODRÍGUEZ; GUILLERMO CORTIÑAS

JOURNAL OF OPERATOR THEORY

THETA FOUNDATION

Lugar: Bucharest; Año: 2019 vol. 81 p. 225 - 254

0379-4024

 For each 1⩽p<∞ and each countable oriented graph Q we introduce an Lp-operator algebra Op(Q), which contains the Leavitt path C-algebraLQ as a dense subalgebra, and is universal for those Lp-representations of LQ which are spatial in the sense of N.C. Phillips. We prove that Op(Q) is simple as an Lp-operator algebra if and only if LQ is simple, in which case it is isometrically isomorphic to ¯¯¯¯¯¯¯¯¯¯¯¯¯¯ρ(LQ) for any nonzero spatial Lp-representation ρ:LQ→L(Lp(X)). If moreover LQ is purely infinite simple and p≠p′, then there is no nonzero continuous homomorphism Op(Q)→Op′(Q).