Fixed-point Cuntz semigroup as a sub-Cu-semigroup (general actions)

Determine whether, for an arbitrary action α: G → Aut(A) of a discrete group G on a C*-algebra A, the fixed-point submonoid Cu(A)^α = {x ∈ Cu(A): Cu(α_g)(x) = x for all g ∈ G} is a sub-Cu-semigroup of Cu(A) (i.e., the inclusion Cu(A)^α → Cu(A) is a Cu-morphism and Cu(A)^α satisfies the Cu-axioms), beyond the cases covered by Proposition 3.29(2) where α has the weak tracial Rokhlin property.

Background

The paper defines the fixed-point semigroup Cu(A)^α associated to an action α: G → Aut(A) on a C*-algebra A by requiring invariance under the induced action on Cu(A). It is observed that Cu(A)^α is a submonoid of Cu(A) and is closed under suprema of increasing sequences, but it is not generally known whether it satisfies all Cu-axioms and whether the inclusion is a Cu-morphism.

The authors prove that in the specific setting of finite group actions with the weak tracial Rokhlin property on stably finite simple non-type-I C*-algebras, Cu(A)^α is indeed a sub-Cu-semigroup (Proposition 3.29(2)). The open problem asks for the general case without those assumptions.