A notion of fractality for a class of states and noncommutative relative distance zeta functional

TL;DR

This paper extends the concept of relative distance zeta functions to states over C* algebras, introducing a new framework to analyze fractality and complex dimensions in noncommutative geometry.

Contribution

It proposes a novel relative distance zeta functional for states on C* algebras, generalizing previous set-based notions and enabling fractality analysis in noncommutative spaces.

Findings

Defined a new relative distance zeta functional for C* algebra states

Explored properties and decomposition rules of the zeta functional

Identified fractal properties and complex dimensions in examples

Abstract

In this work, we first recall the definition of the relative distance zeta function in [42, 43, 44, 46, 47] and slightly generalize this notion from sets to probability measures, and then move on to propose a novel definition a relative distance (and tube) zeta functional for a class of states over a C* algebra. With such an extension, we look into the chance to define relative Minkowski dimensions in this context, and explore the notion of fractality for this class of states. Relative complex dimensions as poles of this newly proposed relative distance zeta functional, as well as its geometric and transformation properties, decomposition rules and properties that respects tensor products are discussed. We then explore some examples that possess fractal properties with this new zeta functional and propose functional equations similar to [11,35,36,42].