Examples of infinite direct sums of spectral triples

TL;DR

This paper explores methods for summing infinite families of noncommutative spectral triples, introducing a new integration concept and applying it to Toeplitz operator algebras on complex spaces, including fractals.

Contribution

It proposes a novel definition of spectral triple integration and demonstrates its application to Toeplitz algebras on complex domains and fractals, linking spectral triples to geometric properties.

Findings

Defined a new integration method for spectral triples

Constructed spectral triples for Toeplitz algebras on complex spaces

Recovered Hausdorff dimension of fractal sets using spectral triples

Abstract

We study two ways of summing an infinite family of noncommutative spectral triples. First, we propose a definition of the integration of spectral triples and give an example using algebras of Toeplitz operators acting on weighted Bergman spaces over the unit ball of $\mathbb{C}^n$. Secondly, we construct a spectral triple associated to a general polygonal self-similar set in $\mathbb{C}$ using algebras of Toeplitz operators on Hardy spaces. In this case, we show that we can recover the Hausdorff dimension of the fractal set.