Quantum Automorphism Group of Direct Sum of Cuntz Algebras

In 1998, Wang constructed an ergodic action of the compact quantum group Uₙ⁺ (free unitary quantum group) on the Cuntz algebra Oₙ. Later, in 2018-2019, S. Joardar and A. Mandal showed that the quantum automorphism group of the Cuntz algebra Oₙ (as a graph C^*-algebra) is Uₙ⁺ in the category introduced by them. In this article, we explore the quantum symmetry of the direct sum of Cuntz algebras viewing them as a graph C^*-algebra in the category as mentioned before. It has been shown that the quantum automorphism group of the direct sum of non-isomorphic Cuntz algebras ᵢ\ᵢ₌₁ᵐ is Un₁⁺*Un₂⁺* ⋯ *Unₘ⁺ for distinct nᵢ's, i.e. if Lnᵢ (the graph contains nᵢ loops based at a single vertex) is the underlying graph of Onᵢ, then {equation*} QτLin(ᵢ₌₁ᵐ ~ Ln_i) *ᵢ₌₁ᵐ ~~ QτLin(Ln_i) {U}n_1⁺*{U}n_2⁺* ⋯ *{U}n_m⁺. {equation*} Moreover, the quantum symmetry of the direct sum of m copies of isomorphic Cuntz algebra Oₙ (whose underlying graph is Lₙ) is Uₙ⁺ _* Sₘ⁺, i.e. {equation*} QτLin(ᵢ₌₁ᵐ ~ L_n) QτLin(L_n) _* S_m^+ U_n^+ _* S_m^+. {equation*} On the other hand, it is known that the quantum automorphism group of m disjoint copies of a simple, connected graph Γ is isomorphic to the free wreath product of the quantum automorphism group of Γ with Sₘ⁺. Though an analoguous result is true for Oₙ (as a graph C^*-algebra), we have provided a counter-example to show that this result is not in general true for an arbitrary graph C^*-algebra.