2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/123205Let $\ell$ 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.26 pages. Various small improvements. Two references addedOperator AlgebrasGroup TheoryMetric Geometry46L87; 20F67, 46L09text