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

Published:2005-10-01
Printed: Oct 2005
• Narutaka Ozawa
• Marc A. Rieffel
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## Abstract

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 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.
 MSC Classifications: 46L87 - Noncommutative differential geometry [See also 58B32, 58B34, 58J22] 20F67 - Hyperbolic groups and nonpositively curved groups 46L09 - Free products of $C^*$-algebras