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Simplicity of Cuntz-Pimsner algebras of quantum graphs

Mitch Hamidi, Lara Ismert, Brent Nelson

TL;DR

The paper develops a detailed framework to study the simplicity of Cuntz–Pimsner algebras arising from quantum graphs via their quantum edge correspondences. It connects quantum Cuntz–Krieger algebras to relative Cuntz–Pimsner algebras through explicit surjections and analyzes when these algebras are (non-)simple, using non-returning vectors and Condition (S) to certify simplicity in non-full cases. A key contribution is a concrete example showing ${\mathcal{O}}({\mathcal{G}})$ and ${\mathcal{O}}_{E_{\mathcal{G}}}$ can be non-isomorphic, even when the quantum edge correspondence is faithful but not full. The authors derive explicit simplicity criteria in several regimes: single-vertex graphs via Kraus-dimension (Marrero–Muhly), complete graphs, and trivial graphs, and they formulate a Schweizer-based minimality/aperiodicity criterion for the general full-case. Collectively, the results advance understanding of when quantum graph algebras are simple and how the Cuntz–Krieger and Cuntz–Pimsner perspectives diverge.

Abstract

Let $\mathcal{G}$ be a quantum graph without quantum sources and $E_\mathcal{G}$ be the quantum edge correspondence for $\mathcal{G}.$ Our main results include sufficient conditions for simplicity of the Cuntz-Pimsner algebra $\mathcal{O}_{E_\mathcal{G}}$ in terms of $\mathcal{G}$ and for defining a surjection from the quantum Cuntz-Krieger algebra $\mathcal{O}(\mathcal{G})$ onto a particular relative Cuntz-Pimsner algebra for $E_\mathcal{G}$. As an application of these two results, we give the first example of a quantum graph with distinct quantum Cuntz-Krieger and local quantum Cuntz-Krieger algebras. We also characterize simplicity of $\mathcal{O}_{E_\mathcal{G}}$ for some fundamental examples of quantum graphs, including rank-one quantum graphs on a single full matrix algebra, complete quantum graphs, and trivial quantum graphs. Along the way, we provide an equivalent condition for minimality of $E_\mathcal{G}$ and sufficient conditions for aperiodicity of $E_\mathcal{G}$ in terms of the underlying quantum graph $\mathcal{G}$.

Simplicity of Cuntz-Pimsner algebras of quantum graphs

TL;DR

The paper develops a detailed framework to study the simplicity of Cuntz–Pimsner algebras arising from quantum graphs via their quantum edge correspondences. It connects quantum Cuntz–Krieger algebras to relative Cuntz–Pimsner algebras through explicit surjections and analyzes when these algebras are (non-)simple, using non-returning vectors and Condition (S) to certify simplicity in non-full cases. A key contribution is a concrete example showing and can be non-isomorphic, even when the quantum edge correspondence is faithful but not full. The authors derive explicit simplicity criteria in several regimes: single-vertex graphs via Kraus-dimension (Marrero–Muhly), complete graphs, and trivial graphs, and they formulate a Schweizer-based minimality/aperiodicity criterion for the general full-case. Collectively, the results advance understanding of when quantum graph algebras are simple and how the Cuntz–Krieger and Cuntz–Pimsner perspectives diverge.

Abstract

Let be a quantum graph without quantum sources and be the quantum edge correspondence for Our main results include sufficient conditions for simplicity of the Cuntz-Pimsner algebra in terms of and for defining a surjection from the quantum Cuntz-Krieger algebra onto a particular relative Cuntz-Pimsner algebra for . As an application of these two results, we give the first example of a quantum graph with distinct quantum Cuntz-Krieger and local quantum Cuntz-Krieger algebras. We also characterize simplicity of for some fundamental examples of quantum graphs, including rank-one quantum graphs on a single full matrix algebra, complete quantum graphs, and trivial quantum graphs. Along the way, we provide an equivalent condition for minimality of and sufficient conditions for aperiodicity of in terms of the underlying quantum graph .

Paper Structure

This paper contains 10 sections, 25 theorems, 43 equations.

Table of Contents

  1. Preliminaries
  2. Quantum graphs and quantum edge correspondences
  3. Quantum Cuntz--Krieger algebras
  4. Relative Cuntz--Pimsner algebras
  5. QCK ${\mathcal{G}}$-families from Toeplitz $E_{\mathcal{G}}$-representations
  6. Non-returning vectors and Condition (S)
  7. A quantum graph ${\mathcal{G}}$ such that ${\mathcal{O}}({\mathcal{G}})$ is not isomorphic to ${\mathcal{O}}_{E_{\mathcal{G}}}$
  8. Simplicity of ${\mathcal{O}}_{E_{\mathcal{G}}}$
  9. Simplicity of ${\mathcal{O}}_{E_{\mathcal{G}}}$ when $B$ is a single full matrix algebra
  10. Simplicity of ${\mathcal{O}}_{E_{\mathcal{G}}}$ in terms of minimality and periodicity of $E_{\mathcal{G}}$

Key Result

Theorem 1.1

Let ${\mathcal{G}}=(B,\psi,A)$ be a quantum graph such that $\psi$ is a $\delta$-form and $A$ is cp, and let $E_{\mathcal{G}}$ be the quantum edge correspondence. Then and the two-sided ideal of $B$ generated by the range of $A$. In particular, $E_{\mathcal{G}}$ is faithful if and only if $\ker(A)$ does not contain a central summand of $B$, and $E_{\mathcal{G}}$ is full if and only if $A(B)$ is

Theorems & Definitions (35)

  • Theorem 1.1: BHINW23
  • Definition 1.2: BEVW22,BHINW23
  • Definition 1.3: BHINW23
  • Theorem 1.4: BHINW23
  • Lemma 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Remark 1.8
  • Theorem 2.1: BHINW23
  • Theorem 2.2: BHINW23
  • ...and 25 more