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

TL;DR

This paper explores the structure of $C^*$-algebras linked to mirror quantum spheres, demonstrating their isomorphism to Cuntz-Pimsner algebras and labelled graph $C^*$-algebras, and establishing a functorial relationship between labelled graphs and $C^*$-correspondences.

Contribution

It introduces a functor from labelled graphs to $C^*$-correspondences and shows that $C^*$-algebras of mirror quantum spheres can be described via this framework.

Findings

$C^*$-algebras of mirror quantum spheres are isomorphic to Cuntz-Pimsner algebras.

A functor from labelled graphs to $C^*$-correspondences is constructed.

$C^*$-correspondences for these spheres arise from a general restricted direct sum construction.

Abstract

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.