Quantum automorphism groups of direct sums of Cuntz algebras

We explore the quantum symmetry of the direct sum of a finite family {Oni}mi=1 of Cuntz algebras, viewing them as graph C∗-algebras associated to the graphs {Lni}mi=1 (where Ln 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+n1∗⋯∗U+nm for distinct ni’s, i.e. QLinτ(⨆i=1mLni)≅∗i=1mQLinτ(Lni)≅U+n1∗⋯∗U+nm, where QLinτ(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 On is U+n≀∗S+m, i.e. QLinτ(⨆i=1mLn)≅QLinτ(Ln)≀∗S+m≅U+n≀∗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.