Group C^*-algebras as compact quantum metric spaces

Let be a length function on a group G , and let M_{} denote the operator of pointwise multiplication by on ^2(G) . Following Connes, M_{} 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 = { Z}^d when is a word-length, or the restriction to { Z}^d of a norm on { 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.