A universal coefficient theorem for actions of finite groups on C*-algebras

The equivariant bootstrap class in the Kasparov category of actions of a finite group G consists of those actions that are equivalent to one on a Type I C*-algebra. Using a result by Arano and Kubota, we show that this bootstrap class is already generated by the continuous functions on G/H for all cyclic subgroups H of G. Then we prove a Universal Coefficient Theorem for the localisation of this bootstrap class at the group order |G|. This allows us to classify certain G-actions on stable Kirchberg algebras up to cocycle conjugacy.

💡 Research Summary

The paper establishes a universal coefficient theorem (UCT) for actions of finite groups on C*-algebras and applies it to the classification of outer actions on stable Kirchberg algebras up to cocycle conjugacy. The authors begin by defining the equivariant bootstrap class inside the Kasparov category KK^G as the localising subcategory generated by actions that are KK‑equivalent to a Type I C*-algebra. Using a recent result of Arano and Kubota, they show that this bootstrap class is already generated by the elementary objects C(G/H) for all cyclic subgroups H ⊂ G. In other words, any object in the bootstrap class can be built from induced algebras Ind_G^H C(G/H) and triangulated constructions, because the restriction functors to all cyclic subgroups detect zero objects.

The authors then introduce a homological ideal I given by the intersection of the kernels of the restriction functors Res_H^G for all cyclic H. Objects of the form Ind_G^H A (with A in KK^H) are I‑projective, and there are enough such projectives to construct an I‑projective resolution of any bootstrap object. By forming the bar resolution associated to the comonad T = ⊕_{H cyclic} Ind_G^H Res_H^G, they obtain a concrete I‑projective resolution. The “phantom castle” technique shows that the homotopy colimit of the I‑cellular approximations recovers the original object, confirming that the bootstrap class coincides with the localising subcategory generated by the C(G/H) for cyclic H.

Next, the paper studies the Mackey module structure on KK^G(C(G/H), B). Dell’Ambrogio’s work shows that these groups form a Z/2‑graded Mackey functor k_G^(B) over the representation Green ring R_G. For each cyclic H with order n, the authors define a submodule F_H^(B) consisting of elements killed by the n‑th cyclotomic polynomial Φ_n(z). This submodule is naturally a module over the smash product ring ℤ