Nilpotent Group C*-algebras as Compact Quantum Metric Spaces

Christ, Michael ; Rieòel, Marc A.

Canadian mathematical bulletin, Tome 60 (2017) no. 1, pp. 77-94

Résumé

Let $\mathbb{L}$ be a length function on a group $G$ , and let ${{M}_{\mathbb{L}}}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on ${{\ell }^{2}}\left( G \right)$ . Following Connes, ${{M}_{\mathbb{L}}}$ can be used as a “Dirac” operator for the reduced group ${{C}^{*}}$ -algebra $C_{r}^{*}\left( G \right)$ . It defines a Lipschitz seminorm on $C_{r}^{*}\left( G \right)$ , which defines a metric on the state space of $C_{r}^{*}\left( G \right)$ . We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak- $*$ topology (a key property for the definition of a “compact quantum metric space”). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

Détail

DOI : 10.4153/CMB-2016-040-6

Mots-clés : 46L87, 20F65, 22D15, 53C23, 58B34, group C*-algebra, Dirac operator, quantummetric space, discrete nilpotent group, polynomial growth