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

Abstract

Let $\ell$ 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 $C_r^{\ast }(G)$. It defines a Lipschitz seminorm on $C_r^{\ast }(G)$, which defines a metric on the state space of $C_r^{\ast }(G)$. 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.

26 pages. Various small improvements. Two references added

References in corpus (3)

• Metrics on State Spaces
• Metrics on states from actions of compact groups
• Group C*-algebras, metrics and an operator theoretic inequality

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