Hyperbolic group $C^*$-algebras and free-product $C^*$-algebras as compact quantum metric spaces

Details Hyperbolic group $C^*$-algebras and free-product $C^*$-algebras as compact quantum metric spaces Hyperbolic group $C^*$-algebras and free-product $C^*$-algebras as compact quantum metric spaces

2003-02-25

arxiv.org/abs/math/0302310v3

Canad. J. Math. 57 (2005) 1056-1079

Mathematics

Operator Algebras

26 pages. Various small improvements. Two references added

Scientific paper

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 $C_r^*(G)$. 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.

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