Group -algebras as compact quantum metric spaces

  • Marc A. Rieffel

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

This article is published open access.


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. 605–651

DOI 10.4171/DM/133