2002/05/31 by Marc A. Rieffel
Mathematics · #math.OA #math.GR #math.MG #msc:47L87 #msc:20F65 #msc:53C23 #msc:58B34
published as Documenta Mathematica 7 (2002) 605-651 · 53 pages, yet more minor improvements. To appear in Doc. Math
arxiv created 2002/12/19 · arxiv updated 2009/11/30
Let ℓ be a length function on a group G, and let Mℓ denote the operator of pointwise multiplication by ℓ on \bell2(G). Following Connes, Mℓ can be used as a ``Dirac'' operator for Cr^*(G). It defines a Lipschitz seminorm on Cr^*(G), which defines a metric on the state space of Cr^*(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 = \mathbb Zd when ℓ is a word-length, or the restriction to \mathbb Zd of a norm on \mathbb Rd. This works for Cr^*(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.