# Group $C_{∗}$-algebras as compact quantum metric spaces

### Marc A. Rieffel

## Abstract

Let $ℓ$ be a length function on a group $G$, and let $M_{ℓ}$ denote the operator of pointwise multiplication by $ℓ$ on $ℓ_{2}(G)$. Following Connes, $M_{ℓ}$ 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 $ℓ$ 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.

## Cite this article

Marc A. Rieffel, Group $C_{∗}$-algebras as compact quantum metric spaces. Doc. Math. 7 (2002), pp. 605–651

DOI 10.4171/DM/133