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On amplified graph C*-algebras as cores of Cuntz-Krieger algebras

Francesco D'Andrea, Sophie Emma Zegers

TL;DR

This work proves that the AF core of a unital Cuntz–Krieger algebra can be realized as the fixed-point subalgebra under a gauge U(1) action of an amplified graph C*-algebra associated to a finite directed acyclic graph R, via C^*(F_R) ≅ C^*(E_R)^{U(1)}. It derives the dimension-group-based proof of the main isomorphism and applies it to quantum homogeneous spaces, yielding AF presentations for C_q(Gr(2,4)), CP^{n-1}_q, quantum teardrops, and related circle bundles, while clarifying nontrivial K-theory distinctions between classical and quantum projective spaces. Moreover, the paper develops a CW-type (strict) decomposition for Gr_q(2,4) by assembling skeleta through pullback diagrams and Toeplitz-operator–based “cells,” illustrating how noncommutative geometry mirrors classical cell attachments. These results provide structural and computational tools for K-theory and/noncommutative topology in quantum homogeneous spaces, highlighting how amplified graphs encode the topological skeletons of quantum spaces.

Abstract

Given a finite directed acyclic graph $R$, we construct from it two graphs $E_R$ and $F_R$, one by adding a loop at every vertex of $R$ and one by replacing every arrow of $R$ by countably infinitely many arrows. We show that the graph C*-algebra $C^*(F_R)$ is isomorphic to the AF core of $C^*(E_R)$. Examples include C*-algebras of a quantum flag manifolds and quantum teardrops. We discuss in detail the quantum Grassmannian $Gr_q(2,4)$ and use our description as AF core to study its CW-structure.

On amplified graph C*-algebras as cores of Cuntz-Krieger algebras

TL;DR

This work proves that the AF core of a unital Cuntz–Krieger algebra can be realized as the fixed-point subalgebra under a gauge U(1) action of an amplified graph C*-algebra associated to a finite directed acyclic graph R, via C^*(F_R) ≅ C^*(E_R)^{U(1)}. It derives the dimension-group-based proof of the main isomorphism and applies it to quantum homogeneous spaces, yielding AF presentations for C_q(Gr(2,4)), CP^{n-1}_q, quantum teardrops, and related circle bundles, while clarifying nontrivial K-theory distinctions between classical and quantum projective spaces. Moreover, the paper develops a CW-type (strict) decomposition for Gr_q(2,4) by assembling skeleta through pullback diagrams and Toeplitz-operator–based “cells,” illustrating how noncommutative geometry mirrors classical cell attachments. These results provide structural and computational tools for K-theory and/noncommutative topology in quantum homogeneous spaces, highlighting how amplified graphs encode the topological skeletons of quantum spaces.

Abstract

Given a finite directed acyclic graph , we construct from it two graphs and , one by adding a loop at every vertex of and one by replacing every arrow of by countably infinitely many arrows. We show that the graph C*-algebra is isomorphic to the AF core of . Examples include C*-algebras of a quantum flag manifolds and quantum teardrops. We discuss in detail the quantum Grassmannian and use our description as AF core to study its CW-structure.
Paper Structure (14 sections, 18 theorems, 84 equations, 7 figures, 1 table)

This paper contains 14 sections, 18 theorems, 84 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph C*-algebras
  4. The dimension group
  5. The main theorem
  6. Applications
  7. The dimension group of $\mathbb{C} P^{n-1}_q$
  8. Quantum lens spaces and teardrops
  9. Circle bundles over quantum flag manifolds
  10. A CW-type decomposition of $Gr_q(2,4)$
  11. The classical case
  12. Some auxiliary quantum spaces
  13. The CW-structure
  14. On the dimension group of a complex projective space

Key Result

Theorem 3.2

Let $R\subseteq V\times V$ be a finite directed acyclic graph. Then where the graphs $E_R$ and $F_R$ are defined in eq:E and eq:F, respectively.

Figures (7)

  • Figure 1: The graph $L_{2n-1}$.
  • Figure 3: The graph $F_{n-1}$.
  • Figure 5: The graph $L_3^{r;1,r}$.
  • Figure 7: The graph $\widetilde{L}_{2,4}$.
  • Figure 9: The graph $L_{2,4}$.
  • ...and 2 more figures

Theorems & Definitions (36)

  • Example 3.1
  • Theorem 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Corollary 4.1
  • Corollary 4.2
  • proof
  • Proposition 4.3
  • ...and 26 more