The Noncommutative Geometry of the Discrete Heisenberg Group

Motivated by the search for new examples of “noncommutative manifolds”, we study the noncommutative geometry of the group C*-algebra of the three dimensional discrete Heisenberg group. We present a unified treatment of the K-homology, cyclic cohomology and derivations of this algebra. 1. The discrete Heisenberg group Recently, there has been considerable interest in the notion of a “noncommutative manifold”. There are several different approaches to this problem. The first is the operatoralgebraic approach pioneered by Connes in [Co96], which takes as its starting point the commutative algebra C(M), where M is a compact Riemannian spin manifold, together with the Dirac operator coming from the spin connection, and then gives an axiomatic formulation which extends to noncommutative algebras. The most accessible example of a “noncommutative differentiable manifold” in this sense is provided by the noncommutative tori [Ri81]. Recently Connes’ program has enjoyed considerable success [CD01], [CL00], [Var01]. A second approach, via quantum groups, has been much more examples driven, proceeding with case by case studies of the many interesting algebras arising from quantum groups [Ma00], [Sch99], [Wo87]. Rather than constructing spectral triples, these workers focus on classifying the possible differential calculi over a given algebra. The route we take in this paper falls between these two approaches. We start with a specific example, and see how much of Connes’ formalism is applicable. In Connes’ picture, a noncommutative geometry over a C*-algebra A consists of a spectral triple over A equipped with additional structures. The bounded formulation of spectral triples are Fredholm modules, equivalence classes of which make up the Kasparov K-homology groups KK(A,C) (i = 0, 1). Thus the focus of this work is to calculate the K-homology groups and exhibit the generating Fredholm modules. We will not at this point address the problem of finding corresponding spectral triples, and then trying to equip these with the additional structures of a noncommutative geometry. We will study in detail the group C*-algebra of the discrete Heisenberg group H3. This algebra is closely related to the noncommutative tori Aθ it arises as a continuous field of noncommutative tori over the circle and many of the constructions used for the Aθ can also be applied here. However, not everything generalizes so nicely. The group C*-algebra C(H3) is a crossed product C(T )×α Z, and provided the main motivation for our work on K-homology of crossed products by Z [Ha01]. The discrete three-dimensional Heisenberg group H3 can be defined abstractly as the group generated by elements a and b such that the commutator c = abab is central. It can be Date: February 1, 2008. 1991 Mathematics Subject Classification. Primary 58B34; Secondary 19K33, 46L. 1