The structure of the $C^*$-algebras corresponding to even-dimensional mirror quantum spheres is investigated. It is shown that they are isomorphic to both Cuntz-Pimsner algebras of certain $C^*$-correspondences and $C^*$-algebras of certain labelled graphs. In order to achieve this, categories of labelled graphs and $C^*$-correspondences are studied. A functor from labelled graphs to $C^*$-correspondences is constructed, such that the corresponding associated $C^*$-algebras are isomorphic. Furthermore, it is shown that $C^*$-correspondences for the mirror quantum spheres arise via a general construction of restricted direct sum.
David Robertson. Wojciech Szymański. "C∗-algebras associated to C∗-correspondences and applications to mirror quantum spheres." Illinois J. Math. 55 (3) 845 - 870, Fall 2011. https://doi.org/10.1215/ijm/1369841788