Equality in the approximately inner radius inequality

Determine whether the inequality rc(A,alpha)=rc'(A,alpha)=rc(A)≤rc(C*(G,A,alpha)) is necessarily an equality at the crossed-product level for finite-group actions satisfying strict approximate innerness or weak tracial strict approximate innerness.

Background

Theorem \ref{Prp_Rep_1} shows that the ordinary, dynamical, and auxiliary radii agree and are bounded above by the radius of the crossed product for the stated approximately inner action classes.

For the regular-representation model, equality holds. The paper leaves open whether the crossed-product upper bound is always attained under the broader hypotheses of the theorem.