Isomorphism classes for quantum Heisenberg manifolds

We embed the quantum Heisenberg manifold in a crossed product algebra. This enables us to show that, in the irrational case, all tracial states on $\text{dc}$ induce the same homomorphism on the K_0-group. We conclude that two irrational quantum Heisenberg manifolds $\text{dc}$ and $D_{{\mu }^{\mathrm{\prime }}{\nu }^{\mathrm{\prime }}}^c$ are isomorphic if and only if the parameters $(\mu ,\nu )$ and $({\mu }^{\mathrm{\prime }},{\nu }^{\mathrm{\prime }})$ belong to the same orbit under the usual action of $G{L}_2(\text{ZZ})$ on the torus.