Nilpotent Group C*-algebras as Compact Quantum Metric Spaces

By Michael Christ, Marc A. Rieòel. Canadian mathematical bulletin, 2002

75 citations recorded by the source corpus.

Abstract 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a02d56d0ed9f95df648c3a888db753e5fa0884ca