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

TL;DR

This paper constructs spectral triples for AF C*-algebras, including the Cantor set, to induce metrics compatible with their topologies, revealing new geometric insights and classical interpretations.

Contribution

It introduces a natural Dirac operator construction for AF C*-algebras that induces meaningful metrics and connects to classical geometric structures.

Findings

Dirac operators induce metrics on AF C*-algebras' state spaces

Application to the Cantor set yields new Hausdorff dimension representations

Transposition on matrices yields spectral triples with exact norm metric

Abstract

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. In the particular case of a UHF C*-algebra, the construction can be made in a way, which relates directly to the dimensions of the increasing sequence of subalgebras.The algebra of continuous functions on the Cantor set is an approximately finite dimensional C*-algebra and our investigations show, when applied to this algebra, that the proposed Dirac operators have good classical interpretations and lead to an, apparently, new way of constructing a representative for a Cantor set of any given Hausdorff dimension. At the end of the paper we study the finite dimensional full matrix algebras over the complex numbers, and show that the operation of transposition on matrices yields a spectral triple which has the property that it's metric on the state space is exactly the norm distance.This result is then generalized to arbitrary unital C*-algebras.