The Rank Ratio Function and the Radius of Comparison

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.