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

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