Hyperbolic group C*-algebras and free-product C*-algebras as compact quantum metric spaces
N Ozawa, MA Rieffel - Canadian Journal of Mathematics, 2005 - cambridge.org
Canadian Journal of Mathematics, 2005•cambridge.org
Let 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 …
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 …
Abstract
Let 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
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