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Quantum Automorphism Group of Direct Sum of Cuntz Algebras

Ujjal Karmakar, Arnab Mandal

TL;DR

The article determines the quantum automorphism groups for direct sums of Cuntz algebras by casting them as graph C*-algebras and working within three categorical frameworks (Lin, KMS, and orthogonal filtration). It proves precise identifications: for non-isomorphic components, Q_τ^{Lin}(⊔ L_{n_i}) ≅ * U_{n_i}^+; for isomorphic components, Q_τ^{Lin}(⊔ L_N) ≅ U_N^+ ⨀_* S_K^+; and analogous results hold in the KMS and filtration contexts, with counterexamples showing these equalities do not extend universally to arbitrary graph C*-algebras. The work clarifies when free products and wreath products accurately capture quantum symmetries of direct sums and highlights limitations outside the Cuntz-graph framework. Overall, it connects graph-C*-algebra symmetry to well-known compact quantum groups and wreath products, enriching the understanding of quantum symmetries in noncommutative spaces.

Abstract

In this article, we explore the quantum symmetry of the direct sum of a finite family of Cuntz algebras $\{\mathcal{O}_{n_i} \}_{i=1}^{m}$, viewing them as graph $C^*$-algebras associated to the graphs $\{L_{n_i}\}_{i=1}^{m}$ (where $L_n$ denotes the graph containing $n$ loops based at a single vertex), in the category introduced by Joardar and Mandal. It has been shown that the quantum automorphism group of the direct sum of non-isomorphic Cuntz algebras is ${U}_{n_1}^{+}*{U}_{n_2}^{+}* \cdots *{U}_{n_m}^{+}$ for distinct $n_i$'s, i.e. \begin{equation*} Q_τ^{Lin}(\sqcup_{i=1}^{m} ~ L_{n_i}) \cong *_{i=1}^{m} ~~ Q_τ^{Lin}(L_{n_i}) \cong {U}_{n_1}^{+}*{U}_{n_2}^{+}* \cdots *{U}_{n_m}^{+}, \end{equation*} where $Q_τ^{Lin}(Γ)$ denotes the quantum automorphism group of the graph $C^*$-algebra associated to $Γ$. Also, the quantum automorphism group of the direct sum of $m$ copies of isomorphic Cuntz algebra $\mathcal{O}_n$ is $U_n^+ \wr_* S_m^+$, i.e. \begin{equation*} Q_τ^{Lin}(\sqcup_{i=1}^{m} ~ L_n) \cong Q_τ^{Lin}(L_n) \wr_* S_m^+ \cong U_n^+ \wr_* S_m^+. \end{equation*} Furthermore, we have provided counter-examples to demonstrate that the isomorphisms mentioned above cannot be generalized to arbitrary graph $C^*$-algebras, whereas analogous relations can be extended in the context of quantum automorphism groups of graphs in the sense of Banica and Bichon.

Quantum Automorphism Group of Direct Sum of Cuntz Algebras

TL;DR

The article determines the quantum automorphism groups for direct sums of Cuntz algebras by casting them as graph C*-algebras and working within three categorical frameworks (Lin, KMS, and orthogonal filtration). It proves precise identifications: for non-isomorphic components, Q_τ^{Lin}(⊔ L_{n_i}) ≅ * U_{n_i}^+; for isomorphic components, Q_τ^{Lin}(⊔ L_N) ≅ U_N^+ ⨀_* S_K^+; and analogous results hold in the KMS and filtration contexts, with counterexamples showing these equalities do not extend universally to arbitrary graph C*-algebras. The work clarifies when free products and wreath products accurately capture quantum symmetries of direct sums and highlights limitations outside the Cuntz-graph framework. Overall, it connects graph-C*-algebra symmetry to well-known compact quantum groups and wreath products, enriching the understanding of quantum symmetries in noncommutative spaces.

Abstract

In this article, we explore the quantum symmetry of the direct sum of a finite family of Cuntz algebras , viewing them as graph -algebras associated to the graphs (where denotes the graph containing loops based at a single vertex), in the category introduced by Joardar and Mandal. It has been shown that the quantum automorphism group of the direct sum of non-isomorphic Cuntz algebras is for distinct 's, i.e. \begin{equation*} Q_τ^{Lin}(\sqcup_{i=1}^{m} ~ L_{n_i}) \cong *_{i=1}^{m} ~~ Q_τ^{Lin}(L_{n_i}) \cong {U}_{n_1}^{+}*{U}_{n_2}^{+}* \cdots *{U}_{n_m}^{+}, \end{equation*} where denotes the quantum automorphism group of the graph -algebra associated to . Also, the quantum automorphism group of the direct sum of copies of isomorphic Cuntz algebra is , i.e. \begin{equation*} Q_τ^{Lin}(\sqcup_{i=1}^{m} ~ L_n) \cong Q_τ^{Lin}(L_n) \wr_* S_m^+ \cong U_n^+ \wr_* S_m^+. \end{equation*} Furthermore, we have provided counter-examples to demonstrate that the isomorphisms mentioned above cannot be generalized to arbitrary graph -algebras, whereas analogous relations can be extended in the context of quantum automorphism groups of graphs in the sense of Banica and Bichon.
Paper Structure (18 sections, 16 theorems, 90 equations, 4 figures)

This paper contains 18 sections, 16 theorems, 90 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and conventions
  4. The Cuntz algebra
  5. Direct sum of $C^*$-algebras
  6. A state on direct sum of $C^*$-algebras
  7. Direct sum of graph $C^*$-algebras
  8. KMS State on Graph $C^*$-algebra at critical inverse temperature
  9. Compact quantum groups and quantum automorphism groups
  10. Quantum symmetry in an orthogonal filtration preserving way
  11. Quantum symmetry of a graph $C^{*}$-algebra
  12. Quantum symmetry of a graph $C^{*}$-algebra in the category $\mathfrak{C}_{\tau}^{Lin}$ introduced in Mandal
  13. Quantum symmetry of a graph $C^{*}$-algebra in the KMS state category introduced in Mandalkms
  14. Quantum symmetry of direct sum of Cuntz algebras in a natural state preserving category
  15. Direct sum of non-isomorphic Cuntz algebras
  16. ...and 3 more sections

Key Result

Proposition 2.3

Let $\Gamma=\{V(\Gamma),E(\Gamma),s,r \}$ be a finite, directed graph. For a path $\gamma=e_1e_2...e_n \in E^{< \infty}(\Gamma)$, we define $S_{\gamma}:=S_{e_{1}}S_{e_{2}}...S_{e_{n}}$, where $e_1,...,e_n \in E(\Gamma)$.

Figures (4)

  • Figure 3: $\sqcup_{i=1}^{m} L_{n_i}$
  • Figure 4: $P_1 \sqcup So_2$
  • Figure 5: $\sqcup_{i=1}^{K} L_{N}$
  • Figure 6: $P_1 \sqcup P_1$

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Corollary 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 34 more