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KK^G-theory is very special for compact groups

Establish that for every compact group G, the equivariant Kasparov theory KK^G is very special, namely that the theory can be formulated so that only the very special canonical matrix corner embeddings (of the form (A,α) → (M_n ⊗ A, γ ⊗ α)) are axiomatically declared invertible and the associated M_2-actions are of the tensor-product form on matrix algebras.

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Background

The paper distinguishes between special and very special G-actions on matrix algebras M_n(A) and defines a theory to be very special when only canonical matrix corner embeddings with very special actions are declared invertible. This structure is central for deriving explicit product formulas.

After proving computability results under very special actions in GKG and noting analogous implications for KKG, the author conjectures that KKG-theory itself is very special for compact groups. This would extend the explicit computational framework (e.g., for product formulas) within KKG when G is compact.

References

It is likely that erything transfers to $C*$-algebras and compact groups, so that we conjecture: If $G$ is a compact group, then $KKG$-theory is very special.

— Computing the $K$-homology $K$-theory product in splitexact algebraic $KK$-theory (2508.03477 - Burgstaller, 5 Aug 2025) in Section 8 (Implications for KK-theory)