Functoriality of the equivariant power map on graded KK-theory

Establish whether, for second countable locally compact groups G and F and a finite F-set Ω, the equivariant power map \hat\otimes_Ω: KK_G(A,B) → KK_{G\wr_Ω F}(A^{\hat\otimes Ω}, B^{\hat\otimes Ω}) defines a functor on the category of separable graded G–C*-algebras (i.e., is compatible with Kasparov products and sends identities to identities). Equivalently, determine whether every morphism in the graded Kasparov category KK_G(A,B) can be factored as a composition of *-homomorphisms and their inverses (which reduces, by known results, to verifying this for the element e ∈ KK(C, Ŝ)).

Background

In Section 5, the authors construct an equivariant power map \hat\otimes_Ω on KK-theory associated with wreath products, and prove that its restriction to separable ungraded G–C*-algebras is a functor from KK_G to KK_{G\wr_Ω F}. This construction underpins their transfer of gamma elements and Baum–Connes results to wreath products.

For graded G–C*-algebras, the map is defined on classes but its functoriality remains unproven. The authors note that functoriality in the graded setting would follow if every morphism in KK_G(A,B) (for graded algebras A,B) factors as a composition of *-homomorphisms and inverses; by existing structural results, this reduces to verifying the corresponding property for a single element e in KK(C, Ŝ).