Marc A. Rieffel

摘要

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