Group C*-algebras as compact quantum metric spaces
| dc.creator | Rieffel, Marc A. | |
| dc.date | 2002-05-17 | |
| dc.date | 2002-12-19 | |
| dc.date.accessioned | 2026-07-07T04:48:34Z | |
| dc.date.available | 2026-07-07T04:48:34Z | |
| dc.description | 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. | |
| dc.description | 53 pages, yet more minor improvements. To appear in Doc. Math | |
| dc.identifier | https://arxiv.org/abs/math/0205195 | |
| dc.identifier | http://arxiv.org/abs/math/0205195 | |
| dc.identifier | Documenta Mathematica 7 (2002) 605-651 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/64096 | |
| dc.subject | Operator Algebras | |
| dc.subject | Group Theory | |
| dc.subject | Metric Geometry | |
| dc.subject | 47L87; 20F65, 53C23, 58B34 | |
| dc.title | Group C*-algebras as compact quantum metric spaces | |
| dc.type | text |