The comparison theory of projections is fundamental to the theory of von Neumann algebras, and is the basis for the type classification of factors. A C*-algebra might have few or no projections, in which case their comparison theory tells us little about the structure of the C*-algebra. The appropriate replacement for projections is positive elements. This idea was first introduced by Cuntz with the purpose of studying dimension functions on simple C*-algebras. Later, the appropriate definition of the radius of comparison of C*-algebras, based on the Cuntz semigroup, was introduced by Andrew Toms to study exotic examples of simple amenable C*-algebras that are not $\mathcal{Z}$-stable.

The importance of the Cuntz semigroup has become apparent in work related to the Elliott classification program. It is generally complicated and large. For simple nuclear C*-algebras, the classifiable ones are those whose Cuntz semigroups are easily understood. With the near completion of the Elliott program, nonclassifiable C*-algebras receive more attention and the Cuntz semigroup is the main additional available invariant.

In this talk, we will show that the radii of comparison of a C*-algebra, the crossed product, and the fixed point algebra under an action of a finite group with the weak tracial Rokhlin property are related.

The talk is based on a joint work with Nasser Golestani and N.~Christopher~Phillips.