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.