Quantum Isometry Groups of Cuntz–Krieger Algebras

• The topic demonstrates that quantum isometry groups extend classical isometries by encoding universal D-isometric actions on Cuntz–Krieger algebras via log-Laplacian spectral triples.
• It details the construction of Ariadne quantum groups through partial isometries and magic unitaries that preserve the Perron–Frobenius eigenvector of the underlying matrix.
• The study reveals ergodic quantum actions on noncommutative spaces like O_N and the Cantor set, highlighting significant advances in understanding quantum symmetries.

Quantum isometry groups of Cuntz–Krieger algebras arise at the intersection of noncommutative geometry, operator algebras, and quantum group theory. These groups capture the symmetries of the noncommutative geometry associated to a Cuntz–Krieger algebra, particularly when the algebra is equipped with a spectral triple derived from the log-Laplacian associated to an underlying topological Markov chain. This framework extends the concept of classical isometry groups and uncovers a new family of compact matrix quantum groups, including genuinely quantum symmetries which can act ergodically on noncommutative algebras and classical spaces such as the Cantor set (Freslon et al., 6 Jan 2026).

1. Cuntz–Krieger Algebras and Associated Spectral Triples

A Cuntz–Krieger algebra $O_A$ is constructed from a finite, primitive $0$–$1$ matrix $A$ and is generated by partial isometries $S_1,\dots,S_N$ subject to the relations

$$\sum_{i=1}^N S_i S_i^* = 1, \qquad S_i^*S_i = \sum_{j=1}^N A_{ij} S_j S_j^*, \;\; i=1,\dots,N.$$

The algebra encodes the dynamics of the topological Markov chain $(\Sigma_A, \sigma_A)$, where $\Sigma_A$ is the space of bi-infinite paths allowed by $A$. The system admits a unique KMS state $\tau$ and a canonical spectral triple $(c, L^2(O_A,\tau), D)$, with $D$ constructed from the log-Laplacian on $\Sigma_A$ via a Dirichlet form and a confining length potential. This makes $O_A$ a candidate for a noncommutative manifold (Freslon et al., 6 Jan 2026).

The Hilbert space $L^2(O_A, \tau)$ is naturally isomorphic to $L^2(\Gamma_A, d\mu_{\Gamma_A})$, where $\Gamma_A$ is the Renault–Deaconu groupoid of the shift space and $\mu_{\Gamma_A}$ an Ahlfors–regular measure. Within each clopen bisection $\Gamma_\gamma$, the log-Laplacian $\Delta$ acts nonlocally and is diagonalized by Haar wavelet bases. The operator $D = -\Delta + V$ is self-adjoint with compact resolvent, satisfying Connes’ requirements for a spectral triple.

2. Quantum Isometry Groups: Definition and Generalities

Given a spectral triple $(\mathcal{A}, H, D)$, a quantum isometry group is a universal compact quantum group $G = \QISO^+(\mathcal{A}, H, D)$ acting on the $C^*$-algebra $A$ in a $D$-isometric manner. Formally, a coaction $\varphi: A \longrightarrow C(G)\otimes A$ is $D$-isometric if:

• $\varphi$ preserves the KMS state,
• the implementing unitary $U$ leaves $\operatorname{Dom}(1\otimes D)$ invariant,
• $[U, 1\otimes D] = 0$.

The category of such actions admits a universal object, i.e., every $D$-isometric action of a compact quantum group factors uniquely through $\QISO^+(\mathcal{A}, H, D)$. This quantum symmetry extends classical isometry group actions, often exhibiting stronger ergodic properties on noncommutative spaces.

3. Ariadne Quantum Groups and Universal Symmetries

The main algebraic construction for quantum isometry groups of Cuntz–Krieger algebras is the hierarchy of Ariadne quantum groups $G_A^\ell$, defined for $\ell \in \mathbb{N} \cup \{\infty\}$:

• The algebra $C(G_A^\ell)$ is generated by partial isometries $u_{\alpha, \beta}$ (multi-indices of words up to length $\ell$), with range and source projections $$p_{\alpha,\beta} = u_{\alpha,\beta}u_{\alpha,\beta}^*$$ and $$q_{\alpha,\beta} = u_{\alpha,\beta}^*u_{\alpha,\beta}$$ forming magic unitaries.
• These projections must preserve the Perron–Frobenius eigenvector of $A$ and satisfy the commutation relations $A p = q A$.
• The coproduct is given by $$\Delta(u_{\alpha,\beta}) = \sum_\gamma u_{\alpha, \gamma} \otimes u_{\gamma, \beta}$$.

For $\ell=\infty$, $G_A^\infty$ is the quantum isometry group $\QISO^+(O_A, L^2(O_A, \tau), D)$. The canonical action

$$\varphi_A: O_A \longrightarrow C(G_A^\infty) \otimes O_A, \quad \varphi_A(S_i) = \sum_{j=1}^N u_{ij} \otimes S_j$$

is universal among $D$-isometric actions (Freslon et al., 6 Jan 2026). The groups $$G_A^1 \supset G_A^2 \supset \cdots \supset G_A^\infty$$ form an infinite decreasing tower in generic situations, with only $G_A^\infty$ providing the universal symmetry.

In the special case when $A$ is the full-ones matrix, the Ariadne quantum group encompasses the unitary easy quantum groups of Mang (Freslon et al., 6 Jan 2026).

4. Ergodic and Faithful Quantum Actions

For the Cuntz algebra $O_N$ (i.e., $A = \mathbf{1}_N$), the quantum isometry group $G_{\mathbf{1}}^\infty$ acts ergodically and faithfully: $$\operatorname{Fix}\, \varphi = \{ x \in O_N : \varphi(x) = 1 \otimes x \} = \mathbb{C} 1.$$ This is in contrast with the classical group $\mathbb{T} \wr S_N$, which never achieves ergodicity on $O_N$. This result underscores the genuinely quantum nature of the symmetries captured by $G_{\mathbf{1}}^\infty$ (Freslon et al., 6 Jan 2026).

Furthermore, by passing to commutative subalgebras and suitable quotients, one obtains compact matrix quantum groups such as $H_{\mathbf{1}}$ acting faithfully and ergodically on the Cantor set, via

$$C(\Sigma_{\mathbf{1}}) \xrightarrow{\;\varphi_H\;} C(H_{\mathbf{1}}) \otimes C(\Sigma_{\mathbf{1}})$$

with $C(\Sigma_{\mathbf{1}})$ the commutative subalgebra corresponding to the infinite path space.

5. Comparison with Related Notions of Quantum Symmetry

The quantum isometry groups derived from log-Laplacians are distinct from those previously defined via preservation of KMS states or orthogonal filtrations. In the approach of Joardar and Mandal (Mandal et al., 2018), the universal quantum group acting via linear maps and preserving the KMS state is characterized by the $A$-commutation (quantum symmetry) relations and unitarity. In particular, for $O_N$, the quantum symmetry group is Wang’s free unitary quantum group $U_n^+$.

In contrast, the log-Laplacian construction for $O_N$ yields $G_{\mathbf{1}}^\infty$, which is strictly larger than $U_n^+$ and admits ergodic actions, a property the latter lacks with respect to the natural spectral triple (Freslon et al., 6 Jan 2026, Joardar et al., 2024). This suggests the log-Laplacian approach uncovers a broader and more robust class of quantum symmetries.

6. Outlook and Open Directions

The explicit computation and structure of the universal quantum isometry group for the log-Laplacian spectral triple on $O_A$ demonstrates a new family of compact quantum groups that interpolate between quantum automorphism groups of graphs and easy quantum groups. The resulting ergodic quantum actions on $O_N$ and the Cantor set reveal quantum symmetries inaccessible to classical or previously-considered quantum automorphism groups.

Open directions include:

• Generalizing to Cuntz–Krieger algebras with more general underlying graphs (e.g., multiple edges, sources).
• Extending the log-Laplacian notion to higher-rank Cuntz--Krieger-type algebras.
• Analyzing further structural properties and representation theory of the Ariadne quantum groups $G_A^\ell$.
• Investigating connections with spectral rigidity and quantum information theory, supported by the ergodicity of these quantum symmetries.

These themes highlight the significant role of quantum isometry groups in advancing the understanding of noncommutative manifolds and their quantum symmetries (Freslon et al., 6 Jan 2026, Joardar et al., 2024, Mandal et al., 2018).