Group $C^*$-algebras as compact quantum metric spaces

Let \ell be a length function on a group G , and let M_{\ell} denote the operator of pointwise multiplication by \ell on \mathbf \ell^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.