  Let $\ell$ be a length function on a group $G$, and let $M_{\ell}$ denote the
operator of pointwise multiplication by $\ell$ on $\bell^2(G)$. Following
Connes, $M_{\ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It
defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state
space of $C_r^*(G)$. We investigate whether the topology from this metric
coincides with the weak-* topology (our definition of a ``compact quantum
metric space''). We give an affirmative answer for $G = {\mathbb Z}^d$ when
$\ell$ is a word-length, or the restriction to ${\mathbb Z}^d$ of a norm on
${\mathbb R}^d$. This works for $C_r^*(G)$ twisted by a 2-cocycle, and thus for
non-commutative tori. Our approach involves Connes' cosphere algebra, and an
interesting compactification of metric spaces which is closely related to
geodesic rays.
