Isomorphism classes for quantum Heisenberg manifolds

Abstract

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 $\dc$ induce the same homomorphism on the K_0-group. We conclude that two irrational quantum Heisenberg manifolds $\dc$ and $D^c_{\mu ' \nu '}$ are isomorphic if and only if the parameters $(\mu,\nu)$ and $(\mu ',\nu ')$ belong to the same orbit under the usual action of $GL_2(\ZZ)$ on the torus.