Let be a length function on a group , and let denote the operator of pointwise multiplication by on . Following Connes, can be used as a “Dirac” operator for . It defines a Lipschitz seminorm on , which defines a metric on the state space of . 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 when is a word-length, or the restriction to of a norm on . This works for 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.
Cite this article
Marc A. Rieffel, Group -algebras as compact quantum metric spaces. Doc. Math. 7 (2002), pp. 605–651DOI 10.4171/DM/133