Hyperbolic group $C^*$-algebras and free-product $C^*$-algebras as compact quantum metric spaces

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dc.creatorOzawa, Narutaka
dc.creatorRieffel, Marc A.
dc.date2003-02-25
dc.date2003-05-13
dc.date.accessioned2026-07-07T07:44:28Z
dc.date.available2026-07-07T07:44:28Z
dc.descriptionLet $\ell$ 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.
dc.description26 pages. Various small improvements. Two references added
dc.identifierhttps://arxiv.org/abs/math/0302310
dc.identifierhttp://arxiv.org/abs/math/0302310
dc.identifierCanad. J. Math. 57 (2005) 1056-1079
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/123205
dc.subjectOperator Algebras
dc.subjectGroup Theory
dc.subjectMetric Geometry
dc.subject46L87; 20F67, 46L09
dc.titleHyperbolic group $C^*$-algebras and free-product $C^*$-algebras as compact quantum metric spaces
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