Sums of two-dimensional spectral triples

Abstract

We study countable sums of two-dimensional modules for the continuous complex functions on a compact metric space and show that it is possible to construct a spectral triple which gives the original metric back. This spectral triple will be finitely summable for any positive parameter. We also construct a sum of two-dimensional modules which reflects some aspects of the topological dimensions of the compact metric space, but this will only give the metric back approximately. At the end we make an explicit computation of the last module for the unit interval in $\mathsf R$. The metric is recovered exactly, the Dixmier trace induces a multiple of the Lebesgue integral and the growth of the number of eigenvalues $N(\Lambda)$ bounded by $\Lambda$ behaves, such that $N(\Lambda)/\Lambda$ is bounded, but without limit for $\Lambda\to \infty$.