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 O₍ (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 \O₍㶁\₈=₁^m is U₍䃑^+*U₍䃒^+* *U₍䂷^+ for distinct nᵢ's, i. e. if L₍㶁 (the graph contains nᵢ loops based at a single vertex) is the underlying graph of O₍㶁, then equation* Q_^Lin (₈=₁^m ~ L₍㶁) *₈=₁^m ~~ Q_^Lin (L₍㶁) U₍䃑^+*U₍䃒^+* *U₍䂷^+. 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* Q_^Lin (₈=₁^m ~ Lₙ) Q_^Lin (Lₙ) _* Sₘ^+ Uₙ^+ _* Sₘ^+. equation* On the other hand, it is known that the quantum automorphism group of m disjoint copies of a simple, connected graph is isomorphic to the free wreath product of the quantum automorphism group of with Sₘ^+. Though an analoguous result is true for O₍ (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.