Quantum Heisenberg manifolds as twisted groupoid C⁎C⁎-algebras

The quantum Heisenberg manifolds are noncommutative manifolds constructed by M. Rieffel as strict deformation quantizations of Heisenberg manifolds and have been studied by various authors. Rieffel constructed the quantum Heisenberg manifolds as the generalized fixed-point algebras of certain crossed product C⁎C⁎-algebras, and they also can be realized as crossed products of C(T2)C(T2) by Hilbert C⁎C⁎-bimodules in the sense of Abadie et al. In this paper, we describe how the quantum Heisenberg manifolds can also be realized as twisted groupoid C⁎C⁎-algebras. In the final section, for a transformation groupoid X×ZX×Z with X   compact, and (E,p,X)(E,p,X) a locally trivial principal G-bundle, where G is a compact abelian group, we form a groupoid Λ that is a locally trivial principal G  -bundle over X×ZX×Z, which reduces to the twist groupoid of the second author when G=TG=T.