2024/02/09 by Karmakar, Ujjal, Mandal, Arnab
#FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2402.06241
In this article, we explore the quantum symmetry of the direct sum of a finite family of Cuntz algebras \Oni \i=1m, viewing them as graph C^*-algebras associated to the graphs \Lni\i=1m (where Ln 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 Un1+*Un2+* ⋯ *Unm+ for distinct ni's, i.e. QτLin(\sqcupi=1m ~ Lni) ≅ *i=1m ~~ QτLin(Lni) ≅ Un1+*Un2+* ⋯ *Unm+, where QτLin(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 Un+ \wr_* Sm+, i.e. QτLin(\sqcupi=1m ~ Ln) ≅ QτLin(Ln) \wr_* Sm+ ≅ Un+ \wr_* Sm+. 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.