Quantum automorphism groups of direct sums of Cuntz algebras
Abstract
We explore the quantum symmetry of the direct sum of a finite family $\{\mathcal O_{n_i} \}_{i=1}^{m}$ of Cuntz algebras, 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 (2018). We show that the quantum automorphism group of the direct sum of non-isomorphic Cuntz algebras is $U_{n_1}^{+}* \cdots *{U}_{n_m}^{+}$ for distinct $n_i$’s, i.e. $$Q_{\tau }^{{\rm Lin}}\Big(\bigsqcup _{i=1}^{m} L_{n_i}\Big) \cong \mathop{*}_{i=1}^{m} Q_{\tau }^{{\rm Lin}}(L_{n_i}) \cong U_{n_1}^{+}* \cdots *U_{n_m}^{+}, $$ where $Q_{\tau }^{{\rm Lin}}(\varGamma )$ denotes the quantum automorphism group of the graph $C^*$-algebra associated to $\varGamma $. 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. $$ Q_{\tau }^{{\rm Lin}}\Big(\bigsqcup _{i=1}^{m} L_n\Big) \cong Q_{\tau }^{{\rm Lin}}(L_n) \wr _* S_m^+ \cong U_n^+ \wr _* S_m^+. $$ Furthermore, we provide counterexamples to demonstrate that the isomorphisms above cannot be generalized to arbitrary graph $C^*$-algebras, whereas analogous relations do extend to quantum automorphism groups of graphs in the sense of Banica and Bichon.
