Group C*-algebras as compact quantum metric spaces

Let $\mathrm{\ell }$ be a length function on a group $G$, and let $M_{\mathrm{\ell }}$ denote the operator of pointwise multiplication by $\mathrm{\ell }$ on ${\text{bell}}^2(G)$. Following Connes, $M_{\mathrm{\ell }}$ 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 $\mathrm{\ell }$ is a word-length, or the restriction to ${\mathbb{Z}}^d$ of a norm on ${\mathbb{R}}^d$. This works for $C_r^{\ast }(G)$ 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.