  While there has been growing interest for noncommutative spaces in recent
times, most examples have been based on the simplest noncommutative algebra:
[x_i,x_j]=i theta_{ij}. Here we present new classes of (non-formal) deformed
products associated to linear Lie algebras of the kind [x_i,x_j]=ic_{ij}^k x_k.
For all possible three-dimensional cases, we define a new star product and
discuss its properties. To complete the analysis of these novel noncommutative
spaces, we introduce noncompact spectral triples, and the concept of star
triple, a specialization of the spectral triple to deformations of the algebra
of functions on a noncompact manifold. We examine the generalization to the
noncompact case of Connes' conditions for noncommutative spin geometries, and,
in the framework of the new star products, we exhibit some candidates for a
Dirac operator. On the technical level, properties of the Moyal multiplier
algebra M(R_\theta^{2n) are elucidated.
