Group $C^*$-algebras as compact quantum metric spaces

Abstract

Let l be a length function on a group G, and let M l denote the operator of pointwise multiplication by l on l 2 (G). Following Connes, M l 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 investigate whether the topology from this metric coincides with the weak-* topology (our definition of a compact quantum metric space). We give an affirmative answer for G = Z d when l is a word-length, or the restriction to Z d of a norm on R d . This works for C* r (G) twisted by a 2-cocycle, and thus for non-commutative tori. Our approach involves Connes' cosphere algebra, and an interesting compactification of metric spaces which is closely related to geodesic rays.