Explicit translation of Kasparov cycles into GK^G level-one morphisms

Demonstrate an explicit translation showing that any Kasparov cycle z = [s_- ⊕ s_+, _- ⊕ _+, T] in KK^G(A,B) can be represented in GK^G as the level-one morphism t_+ ⋅ f_2 ⋅ f_1^{-1} ⋅ Δ_{t_-} ⋅ φ ⋅ e^{-1}, and, furthermore, determine that when the Hilbert modules underlying s_± are built from change of coefficient algebra, external tensor products, and direct sums, the corner embedding e in this translation is very special.

Background

The paper develops GK^G-theory as a generators-and-relations counterpart to KK^G-theory and uses extended double splitexact sequences to describe level-one morphisms. It outlines how explicit formulas enable computation of products when actions are very special.

To extend these computations to KK^G, the author proposes a conjectural translation that presents Kasparov cycles as GK^G level-one morphisms with specified factors (t_+, f_2, f_1^{-1}, Δ{t-}, φ, e^{-1}). If the Hilbert modules in the cycle come from standard constructions (change of coefficients, external tensor products, direct sums), the translation is expected to use a very special corner embedding, enabling the explicit product computation framework to apply.