A Cuntz-Krieger uniqueness theorem for $L^p$-operator graph algebras

Abstract: We continue the study $L^p$-operator algebras associated with directed graphs initiated by Corti~nas and Rodriguez. We establish an $L^p$-analog of the Cuntz-Krieger uniqueness theorem, proving that for a countable graph $Q$ in which every cycle has an entry, a spatial Cuntz-Krieger family in an $L^p$-space generates an injective representation as soon as the idempotents associated to the vertices of $Q$ are nonzero. Additionally, we show that for acyclic graphs, these representations are automatically isometric. While our general approach is inspired by the proofs in the C*-algebra setting, a careful analysis of spatial representations of graphs on $L^p$-spaces is required. In particular, we exploit the interplay between analytical properties of Banach algebras, such as the role of hermitian elements, and geometric notions specific to $L^p$-spaces, such as spatial implementation.