Spectral Triples for nonarchimedean local fields

TL;DR

This paper constructs spectral triples for the C*-algebras of functions on nonarchimedean local fields and their rings of integers, revealing connections to q-hypergeometric functions and analyzing their spectral properties.

Contribution

It introduces a novel spectral triple construction for nonarchimedean local fields using associated trees, linking spectral properties to special functions.

Findings

Spectrum related to roots of q-hypergeometric functions

Constructed spectral triples for both compact and non-compact cases

Analyzed spectral properties and operator behavior

Abstract

Using associated trees, we construct a spectral triple for the C$^*$-algebra of continuous functions on the ring of integers $R$ of a nonarchimedean local field $F$ of characteristic zero, and investigate its properties. Remarkably, the spectrum of the spectral triple operator is closely related to the roots of a $q$-hypergeometric function. We also study a non compact version of this construction for the C$^*$-algebra of continuous functions on $F$, vanishing at infinity.