On split exact sequences and KK-equivalences of amplified graph C*-algebras

We give a general methodology for constructing split exact sequences of amplified graph C*-algebras with sinks. This in turn allows us to construct explicit KK-equivalences with ${\mathbb{C}}^N$ for a large class of C*-algebras, including the quantum Grassmannian ${\mathrm{G}\mathrm{r}}_q(2,4)$. We discuss compatibility with known (quantum) CW-constructions and give an explicit KK-equivalence between the classical and quantum projective spaces $\mathbb{C}{P}^1$ and $\mathbb{C}{P}_q^1$.