Group C*-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 $\bell^2(G)$. 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.

53 pages, yet more minor improvements. To appear in Doc. Math