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

Abstract

AbstractLet ℓ be a length function on a group G, and let Mℓ denote the operator of pointwise multiplication by ℓ on ℓ2(G). Following Connes, Mℓ 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 ℓ 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.