Extend Theorem 4.7 beyond exact C*-algebras

Ascertain whether Theorem 4.7—which asserts that for a simple infinite-dimensional C*-algebra A, a finite group action α: G → Aut(A) with the weak tracial Rokhlin property, p = |G|^{-1}∑_{g∈G}u_g in M((A⊗K)×_{α⊗id}G), any a ∈ Ped(A^α⊗K)\{0}, and any extended trace T ∈ ET(A×α G), one has T(((κ∘λ)(a))p) = (1/|G|)·T((κ∘λ)(a))—remains valid without assuming that A is exact.

Background

Theorem 4.7 establishes a precise scaling of extended traces and dimension functions when passing from a fixed-point element to its image under the averaging projection in the crossed product, under the weak tracial Rokhlin property. The proof uses exactness to identify extended 2-quasitraces with extended traces (EQT2(A×αG) = ET(A×αG)).

Removing exactness would broaden the applicability of subsequent results, including Proposition 4.9 and Theorem 4.11 on relative radii of comparison, to non-exact simple C*-algebras. The authors explicitly pose the question of whether the theorem holds without exactness.