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

Eusebio Gardella, Siri Tinghammar

TL;DR

This work broadens the C*-algebraic uniqueness framework to $L^p$-operator graph algebras by formulating and proving $L^p$-versions of the gauge-invariant and Cuntz–Krieger uniqueness theorems. It develops a spatial $L^p$-CK theory, introduces the gauge action with its spectral subspaces, and establishes a robust AF-structure for acyclic graphs to yield isometric representations. The authors reduce general graphs to countable CK subgraphs via direct limits, enabling faithful representations under suitable hypotheses and expanding the toolkit for $L^p$-operator algebras arising from directed graphs. The results illuminate the interplay between analytic $L^p$-operator properties (like hermitian elements and spatial partial isometries) and graph-theoretic features (such as cycles with entries and acyclicity) in the non-involutive setting.

Abstract

We continue the study of $L^p$-operator algebras associated with directed graphs initiated by Cortiñas and Rodríguez, 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.

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

TL;DR

This work broadens the C*-algebraic uniqueness framework to -operator graph algebras by formulating and proving -versions of the gauge-invariant and Cuntz–Krieger uniqueness theorems. It develops a spatial -CK theory, introduces the gauge action with its spectral subspaces, and establishes a robust AF-structure for acyclic graphs to yield isometric representations. The authors reduce general graphs to countable CK subgraphs via direct limits, enabling faithful representations under suitable hypotheses and expanding the toolkit for -operator algebras arising from directed graphs. The results illuminate the interplay between analytic -operator properties (like hermitian elements and spatial partial isometries) and graph-theoretic features (such as cycles with entries and acyclicity) in the non-involutive setting.

Abstract

We continue the study of -operator algebras associated with directed graphs initiated by Cortiñas and Rodríguez, and we establish -analogs of both the gauge-invariant and the Cuntz-Krieger uniqueness theorems. The first of these asserts that for a graph , a gauge-equivariant spatial representation of its Leavitt path algebra on an -space generates an injective representation whenever the idempotents associated to the vertices of 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 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 -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 -spaces, such as spatial implementation.

Paper Structure

This paper contains 11 sections, 33 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on partial isometries and idempotents
  3. $L^p$-operator graph algebras and Cuntz-Krieger families
  4. Spatial $L^p$-AF-algebras and acyclic graphs
  5. Infinite direct sums of $L^p$-operator algebras
  6. Spatial AF-algebras
  7. Spatial AF-structure in acyclic graphs
  8. The gauge action and its spectral subspaces
  9. Uniqueness theorems for $L^p$-graph algebras
  10. Cuntz-Krieger subgraphs
  11. Uniqueness theorems

Key Result

Theorem A

(Gauge-invariant uniqueness). Let $p\in [1,\infty)$, let $Q$ be a graph, let $A$ be a unitizable $L^p$-operator algebra, and let $(S,T,E)$ be a Cuntz-Krieger $Q$-family in $A$ such that $E_v\neq 0$ for all $v\in Q^0$. Suppose that there is an isometric action $\beta\colon \mathbb{T}\rightarrow \text

Theorems & Definitions (89)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 1
  • Lemma 2.1
  • proof
  • Definition 2
  • Definition 3
  • Definition 4
  • ...and 79 more