Strong Morita Equivalence for Heisenberg C*-algebras and the Positive Cones of Their ^-groups

Introduction. In [14] we began a study of C*-algebras corresponding to projective representations of the discrete Heisenberg group, and classified these C*-algebras up to *-isomorphism. In this sequel to [14] we continue the study of these so-called Heisenberg C*-algebras, first concentrating our study on the strong Morita equivalence classes of these C*-algebras. We recall from [14] that a Heisenberg C* -algebra is said to be of class /, / G {1, 2, 3}, if the range of any normalized trace on its K0 group has rank / as a subgroup of R; results of Curto, Muhly, and Williams [7] on strong Morita equivalence for crossed products along with the methods of [21] and [14] enable us to construct certain strong Morita equivalence bimodules for Heisenberg C*-algebras. For those of class 2 we are able to prove the following: