Local quantum Cuntz--Krieger algebras of dephasing quantum graphs

We establish a general structure theorem for local quantum Cuntz–Krieger families associated to finite quantum graphs whose relations are analogous to those of the partial isometries generating an ordinary graph $C^{\ast }$-algebra, with the quantum edge correspondence governing composition. We apply this structure theorem to the quantum dephasing graph on $M_n$ to obtain an explicit isomorphism between $O_n$ and the local quantum Cuntz–Krieger algebra. More generally, we obtain a unital embedding of $O_n$ in the local quantum Cuntz–Krieger algebra for the dephasing graph on a finite direct sum of $M_n$.