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

Abstract: In this work, we first recall the definition of the relative distance zeta function in [36, 37, 38, 40, 41] 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 [9].