Geodesic
Group $C^*$-algebras as compact quantum metric spaces
Documenta mathematica, Tome 7 (2002), pp. 605-651
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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 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=Zd when l is a word-length, or the restriction to Zd of a norm on 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.
DOI : 10.4171/dm/133
Classification : 20F65, 53C23, 58B34
Mots-clés : Dirac operator, boundary, group C∗-algebra, quantum metric space, metric compactification, geodesic ray, Busemann point 
@article{10_4171_dm_133,
     author = {Marc A. Rieffel},
     title = {Group $C^*$-algebras as compact quantum metric spaces},
     journal = {Documenta mathematica},
     pages = {605--651},
     year = {2002},
     volume = {7},
     doi = {10.4171/dm/133},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/133/}
}
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Marc A. Rieffel. Group $C^*$-algebras as compact quantum metric spaces. Documenta mathematica, Tome 7 (2002), pp. 605-651. doi: 10.4171/dm/133

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