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_{ \tau}^{ Lin}(\sqcup_{ i=1}^{ m} ~ L_{ n_i}) \cong *_{ i=1}^{ m} ~~ Q_{ \tau}^{ 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 $C^*$-algebra associated to $\Gamma$. Also, the quantum automorphism group of the direct sum of $m$ copies of isomorphic Cuntz algebra $\mathcal{ O}_n$ is $U_n^+ \wr_* S_m^+$, i.e. \begin{ equation*} Q_{ \tau}^{ Lin}(\sqcup_{ i=1}^{ m} ~ L_n) \cong Q_{ \tau}^{ 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.
Categories: math.OA