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

TL;DR

This paper extends key uniqueness theorems from C*-algebra graph theory to $L^p$-operator graph algebras, establishing conditions for injectivity and isometry of representations on $L^p$-spaces.

Contribution

It introduces $L^p$-analogs of gauge-invariant and Cuntz-Krieger uniqueness theorems for graph algebras, expanding the theoretical framework to $L^p$-operator algebras.

Findings

Gauge-equivariant spatial representations are injective if vertex idempotents are nonzero.

Representations are automatically isometric for acyclic graphs.

Conditions for injectivity extend to representations where cycles have entries.

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.