Bounds for radii under finite-group actions

Establish whether, for a finite group G acting on a unital C*-algebra A such that both A and C*(G,A,alpha) are infinite dimensional and simple, the inequalities rc(A)/|G|≤rc(A,alpha)≤rc(A) and rc(A)/|G|≤rc'(A,alpha)≤rc(A) hold without additional assumptions on the action.

Background

The paper proves related inequalities under hypotheses such as the weak tracial Rokhlin property, the Rokhlin property, or approximate innerness. It also constructs examples showing that comparison radii can differ.

The unresolved question asks whether the displayed lower and upper bounds remain valid solely from simplicity and infinite dimensionality of the algebra and crossed product.

References

Are the following inequalities valid without any additional assumption on $$?

\frac{rc (A)}{card (G)} \leq rc (A, ) \leq rc (A)

\frac{rc (A)}{card (G)} \leq rc' (A, ) \leq rc (A).

— The Dynamical Radius of Comparison for C*-Dynamical Systems  (2609.30211 - Asadi-Vasfi et al., 24 Sep 2026) in Problem without a label, Section \ref{Sec_4712_Open}

We do not know whether the inequalities in this theorem are necessarily equalities as in Lemma~\ref{L_1412_CGA}.

— The Dynamical Radius of Comparison for C*-Dynamical Systems  (2609.30211 - Asadi-Vasfi et al., 24 Sep 2026) in Immediately after Theorem \ref{WTRPMainCor}, Section \ref{Sec_6827_Rok}