Communications of the
Korean Mathematical Society CKMS

In this paper we give a definition of the Hodge type Laplacian $\Delta$ on a non-commutative manifold which is the smooth dense subalgebra of a $C\sp*$-algebra. We prove that the Laplacian on a quantum Heisenberg manifold is an elliptic operator in the sense that $(\Delta+1)^{-1}$ is compact.

Keywords: quantum Heisenberg manifolds, Laplacian, elliptic operator