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

[摘要] Let $ell$ be a length function on a group $G$, and let $M_{ell}$denote theoperator of pointwise multiplication by $ell$ on $ell^2(G)$.Following Connes,$M_{ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It defines aLipschitz 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 $ell$ isa word-length function on $G$, then the topology from this metriccoincides with theweak-$*$ topology (our definition of a ``compact quantum metricspace''). We show that a convenient framework is that of filtered$C^*$-algebras which satisfy a suitable ``Haagerup-type'' condition. Wealso use thisframework to prove an analogous fact for certain reducedfree products of $C^*$-algebras.

[学科分类] 数学（综合）