Group $C^{\ast }$-algebras as compact quantum metric spaces

Abstract

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 $C_r^{\ast }(G)$. It defines a Lipschitz seminorm on $C_r^{\ast }(G)$, which defines a metric on the state space of $C_r^{\ast }(G)$. 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 $C_r^{\ast }(G)$ 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.