Naturality proof via operator-homotopy picture of KK-theory

Establish a proof of the naturality of the index isomorphism between KK_0(#1 C,B) and K_0(B) that uses only the operator homotopy picture of KK-theory, for general *-homomorphisms B → C that may be degenerate (i.e., not non-degenerate), without relying on equivalence results such as Jensen–Thomsen’s Theorem 2.2.17.

Background

In Section 6, the paper develops the index isomorphism KK_0(#1 C,B) ≅ K_0(B) and proves its naturality with respect to *-homomorphisms B → C. The authors note that while naturality is straightforward for non-degenerate maps, the case of degenerate *-homomorphisms (where the image does not densely span C) is harder to handle within the operator homotopy framework.

To complete their proof in the degenerate case, they invoke a general equivalence result between different pictures of KK-theory (Jensen–Thomsen, Theorem 2.2.17). They explicitly state that they were unable to find a proof using only the operator homotopy picture, leaving open the possibility of a more direct argument within that picture.