The Rank Ratio Function and the Radius of Comparison

TL;DR

This paper introduces the Rank Ratio Function and the Radius of Comparison Function for Cuntz semigroups, providing tools to analyze and construct C*-algebras with specific comparison properties.

Contribution

It defines new functions that control the oscillation of ranks and facilitate the construction of non-classifiable C*-algebras with prescribed radii of comparison.

Findings

The Rank Ratio Function controls the oscillation of ranks in Cuntz semigroups.

The Radius of Comparison Function aids in constructing C*-algebras with specific comparison radii.

New methods for analyzing the relative radii of comparison in Cuntz semigroups.

Abstract

For any given Cuntz semigroup, we introduce a function associated to it, called the ``Rank Ratio Function". This function ensures global control over the oscillation of the rank of any pair of elements in the Cuntz semigroup and, also, plays a very important role in computing the relative radii of comparison of the Cuntz semigroups. Along the way, as we study both the C*-algebraic and algebraic aspects of the relative radius of comparison with a deeper analysis, we introduce another function called the ``Radius of Comparison Function" which provides a method for manufacturing non-classifiable C*-algebras with different relative radii of comparison.