Journal of Operator Theory

Volume 56, Issue 1, Summer 2006 pp. 17-46.

Spectral triples for AF $C^*$-algebras and metrics on the Cantor set

Authors: Erik Christensen (1) and Cristina Ivan
Author institution: (1) Department of Mathematics, University of Copenhagen, Copenhagen , 2100, Denmark
(2) Department of Mathematics, University of Hannover, Hannover, 30167, Germany

Summary: An AF $C^*$-algebra has a natural filtration as an increasing sequence of finite dimensional $C^*$-algebras. We show that it is possible to construct a Dirac operator which relates to this filtration in a natural way and which will induce a metric for the weak*-topology on the state space of the algebra. It turns out that for AF $C^*$-algebras, there is no limit to the growth of the eigenvalues of such a Dirac operator. We have obtained a kind of an inverse to this result, by showing that a phenomenon like this can only occur for AF $C^*$-algebras. The results are then applied to a study of the classical Cantor set.

Keywords: AF $C^*$-algebras, metrics, non commutative compact spaces, spectral triples, Dirac operator, Cantor set, dimension

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