Group $C^*$-Algebras as Compact Quantum Metric Spaces

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.

2000 Mathematics Subject Classification: Primary 47L87; Secondary 20F65, 53C23, 58B34

Keywords and Phrases: Group $C^*$-algebra, Dirac operator, quantum metric space, metric compactification, boundary, geodesic ray, Busemann point.

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