Quantum Automorphism Group of Direct Sum of Cuntz Algebras

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 $C^{\ast }$-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 $U_{n_1}^{+}\ast U_{n_2}^{+}\ast \cdots \ast U_{n_m}^{+}$ 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 $Q_{\tau }^{Lin}(\Gamma )$ denotes the quantum automorphism group of the graph $C^{\ast }$-algebra associated to $\Gamma$. Also, the quantum automorphism group of the direct sum of $m$ copies of isomorphic Cuntz algebra ${\mathcal{O}}_n$ is $U_n^{+}{\wr }_{\ast }{S}_m^{+}$, 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 $C^{\ast }$-algebras, whereas analogous relations can be extended in the context of quantum automorphism groups of graphs in the sense of Banica and Bichon.

The article will appear in Studia Mathematica. A few results have been modified, and some mathematical issues from the previous version have been corrected