Metrics on the Quantum Heisenberg Manifold

For a positive integer c, letHc be the subgroup of G obtained when x, y, cz are integers. The Heisenberg manifold Mc is the quotient G/Hc. Nonzero Poisson brackets on Mc invariant under left translation by G are parametrized by two real parameters μ, ν with μ + ν 6= 0 [4]. For each positive integer c and real numbers μ, ν Rieffel constructed a C*-algebra A μ,ν , quantum Heisenberg manifold (QHM) as example of deformation quantization along a Poisson bracket [4]. These algebras have further been studied by [1] [2] [3] [8]. Recently in a series of papers [5] [6] [7] Rieffel has introduced the notion of compact quantum metric space (CQMS). He has also constructed examples in the case of C*-dynamical systems where the dynamics is driven by a compact Lie group in ergodic way. Now it is also known that Heisenberg group acts ergodically on QHM. Using this action Weaver attempted to produce examples of CQMS out of QHM. His construction does not completely achieve the goal. Here essentially using the technique of Rieffel in a modified way we construct examples of CQMS from QHM. Organization of the paper is as follows. In the next section we briefly recall the notion of QHM and the group action. Then on a suitable dense