Lifting Bratteli Diagrams between Krajewski Diagrams: Spectral Triples, Spectral Actions, and $AF$ algebras

TL;DR

This paper develops a mathematical framework to lift Bratteli diagram arrows to Krajewski diagrams within spectral triples, enabling comparison of spectral actions in non-commutative gauge theories for $AF$-algebras.

Contribution

It introduces a method to construct spectral triples on $AF$-algebras and lifts Bratteli diagram arrows to Krajewski diagrams, enhancing the understanding of non-commutative geometric structures.

Findings

Constructed spectral triples on inductive $AF$-algebras.

Lifted arrows between Bratteli diagrams to Krajewski diagrams.

Compared spectral actions in non-commutative gauge theories.

Abstract

In this paper, we present a framework to construct sequences of spectral triples on top of an inductive sequence defining an $AF$-algebra. One aim of this paper is to lift arrows of a Bratteli diagram to arrows between Krajewski diagrams. The spectral actions defining Non-commutative Gauge Field Theories associated to two spectral triples related by these arrows are compared (tensored by a commutative spectral triple to put us in the context of Almost Commutative manifolds). This paper is a follow up of a previous one in which this program was defined and physically illustrated in the framework of the derivation-based differential calculus, but the present paper focuses more on the mathematical structure without trying to study the physical implications.