Uniqueness theorems for $L^p$-operator graph algebras

Abstract

We continue the study of $L^p$-operator algebras associated with directed graphs initiated by Corti\~nas and Rodr\'iguez, and we establish $L^p$-analogs of both the gauge-invariant and the Cuntz-Krieger uniqueness theorems. The first of these asserts that for a graph $Q$, a gauge-equivariant spatial representation of its Leavitt path algebra $L_Q$ on an $L^p$-space generates an injective representation whenever the idempotents associated to the vertices of $Q$ are nonzero. The second of these theorems states that, in the setting just described, the same conclusion holds if gauge-equivariance is replaced by the assumption that every cycle in $Q$ has an entry. Additionally, we show that for acyclic graphs, such 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.