Simple C*-crossed Product from Discrete 5-dimensional Nilpotent Groups

Abstract

In each of 3 and 4 dimensions there is a unique (up to isomorphism) connected, simply connected, nilpotent Lie group, called G3 and G4, respectively. In 5 dimensions there are 6 such groups G5,i, 1 ≦ i ≦ 6. In [2] operator equations (analogous to UV = λVU for the irrational rotaton algebra Aθ) were used to find cocompact subgroups H5,i ⊂ G5,i that would be analogous to the integer Heisenberg group H3 ⊂ G3. The main thrust in [2] was to identify the infinite dimensional simple quotients of C* (H5,i), both the faithful ones Ai (generated by a faithful representation of H5,i) and the non-faithful ones, and also to give matrix presentations over lower dimensional algebras for as many of the non-faithful quotients as possible. In the course of this work, a small number of the C*-crossed product presentations of the Ai's were mentioned. The purpose of this paper is to display explicitly all the C*-crossed product presentations of the Ai's that are of potential interest and usefulness; they are analogous to C* (C(T), Z) and C*(C, Z2), the flow and cocycle presentations of Aθ.