Hyperbolic group $C^{\ast }$-algebras and free-product $C^{\ast }$-algebras as compact quantum metric spaces

Let $\mathrm{\ell }$ be a length function on a group G, and let $M_{\mathrm{\ell }}$ denote the operator of pointwise multiplication by $\mathrm{\ell }$ on ${\text{bell}}^2(G)$. Following Connes, $M_{\mathrm{\ell }}$ 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 $C^{\ast }$-algebras which satisfy a suitable `` Haagerup-type’’ condition. We also use this framework to prove an analogous fact for certain reduced free products of $C^{\ast }$-algebras.