Hyperbolic Group C *-Algebras and Free-Product C *-Algebras as Compact Quantum Metric Spaces

Ozawa, Narutaka ; Rieffel, Marc A.

Canadian journal of mathematics, Tome 57 (2005) no. 5, pp. 1056-1079

Résumé

Let $\ell $ be a length function on a group $G$ , and let ${{M}_{\ell }}$ denote the operator of pointwise multiplication by $\ell $ on ${{\ell }^{2}}\left( G \right)$ . Following Connes, ${{M}_{\ell }}$ can be used as a “Dirac” operator for $C_{r}^{*}\left( G \right)$ . 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 if $G$ is a hyperbolic group and if $\ell $ is a word-length function on $G$ , then the topology from this metric coincides with the weak- $*$ topology (our definition of a “compact quantum metric space”). We show that a convenient framework is that of filtered ${{C}^{*}}$ -algebras which satisfy a suitable “Haagerup-type” condition. We also use this framework to prove an analogous fact for certain reduced free products of ${{C}^{*}}$ -algebras.

Détail

DOI : 10.4153/CJM-2005-040-0

Mots-clés : Primary: 46L87, secondary: 20F67, 46L09

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