Equality of relative radii of comparison for fixed-point and underlying algebra in a specific Z/2-action

Establish whether, for the stably finite simple unital C*-algebra A equipped with the Z/2Z-action α described in the paper’s Example 5.8 and for any nonzero positive element a ∈ (A^α)+, the equality rc(Cu(A^α), [a]) = rc(Cu(A), [ι(a)]) holds, where ι: A^α → A denotes the inclusion.

Background

Example 5.8 considers a specific stably finite simple unital C*-algebra A with an action α: Z2 → Aut(A) from [7], for which the authors derive an inequality chain relating the radius of comparison of the crossed product to relative radii for the fixed-point algebra and the underlying algebra. In this setting, Theorem 4.11 shows rc(Cu(Aα), [a]) ≤ rc(Cu(A), [ι(a)]).

The open question asks whether these relative radii coincide in this particular example, potentially sharpening the comparison results between the fixed-point algebra and A via the inclusion map.

References

Question 5.9. In Example 5.8, can we prove that rc(Cu(Aa), [a]) = rc(Cu(A), [e(a)])?

— The relative radius of comparison of the crossed product of a non-unital C*-algebra by a finite group  (2505.00952 - Asadi-Vasfi et al., 2 May 2025) in Question 5.9, Section 5

We don't know whether $rc' (A, ) = \frac{1}{4} rc (A_0)$ in general.

— The Dynamical Radius of Comparison for C*-Dynamical Systems  (2609.30211 - Asadi-Vasfi et al., 24 Sep 2026) in Section 3, immediately after Lemma \ref{L_5Z10_PjUn}

In Theorem~\ref{WTRPMainCor}, we will show that some features of this situation generalize to actions satisfying suitable forms of the Rokhlin property. We do not know whether all features of this situation carry over.

— The Dynamical Radius of Comparison for C*-Dynamical Systems  (2609.30211 - Asadi-Vasfi et al., 24 Sep 2026) in Section 3, after Lemma \ref{L_1412_CGA}