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The structure of the $C^*$-algebras corresponding to even-dimensional mirrorquantum spheres is investigated. It is shown that they are isomorphic to bothCuntz-Pimsner algebras of certain $C^*$-correspondences and $C^*$-algebras ofcertain labelled graphs. In order to achieve this, categories of labelledgraphs 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 generalconstruction of restricted direct sum.

$C^*$-algebras associated to $C^*$-correspondences and applications to mirror quantum spheres

$C^$-algebras associated to $C^$-correspondences and applications to mirror quantum spheres