Characterize KKnuc via split exactness with weakly nuclear splitting

Determine an appropriate split exactness property for exact sequences of C*-algebras with weakly nuclear splitting homomorphisms that characterizes the nuclear KK-theory functor KKnuc(A, B), thereby clarifying whether such a characterization holds and specifying the exact formulation of the axiom.

Background

In Section 7, the authors show that the functors KK(X; * , ·) and KKG are characterized—like ordinary KK—by split exactness together with homotopy invariance and stability within their respective categories. They then suggest that an analogous characterization might exist for KKnuc, but would require a more involved split exactness property tailored to sequences with weakly nuclear splittings.

The nuclear KK-theory KKnuc is introduced earlier via the qA formalism and nuclear Kasparov modules, with products and associativity established. However, a universal characterization analogous to KK(X; * , ·) and KKG is not provided, and the authors explicitly leave this question unresolved.

References

It seems that KKnuc could also be characterized by a suitable more involved split exactness property for exact sequences with a weakly nuclear splitting. We leave that open - partly also because we think that such a characterization would be of minor interest.

— Generalized homomorphisms and KK with extra structure  (2404.06840 - Cuntz et al., 2024) in Section 7, Universality and connection to the usual definitions