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

Published online by Cambridge University Press:  20 November 2018

Narutaka Ozawa  and
Marc A. Rieffel
Narutaka Ozawa
Affiliation:
Department of Mathematical Science, University of Tokyo, Komaba, 153-8914, Japan, e-mail: narutaka@ms.u-tokyo.ac.jp
Marc A. Rieffel
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, U.S.A., e-mail: rieffel@math.berkeley.edu
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Abstract

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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.

Keywords

Primary: 46L87secondary: 20F6746L09
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 5 , 01 October 2005 , pp. 1056 - 1079
Copyright
Copyright © Canadian Mathematical Society 2005

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