Group $C^*$-algebras as compact quantum metric spaces

Rieffel, Marc A.

Documenta mathematica, Tome 7 (2002), pp. 605-651.

Résumé

Summary: 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 $C_r^*(G)$. 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 $\ell$ is a word-length, or the restriction to ${\mathbb Z}^d$ of a norm on ${\mathbb R}^d$. 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.

Détail

Classification : 47L87, 20F65, 53C23, 58B34
Keywords: group $C^*$-algebra, Dirac operator, quantum metric space, metric compactification, boundary, geodesic ray, Busemann point