In Kasparov's paper, the canonical isomorphisms $KK_* \rightarrow KK_{*+2k}$ are defined rather implicitely (by tensoring and stabilization).

Are there morphisms of $C^*$-algebras which induce them (e.g. I've heard that the morphism $\varphi: \mathbb{C} \rightarrow \mathbb{C}_2$ sending 1 to $1 + i e_1 e_2$ induces the iso $KK_* \rightarrow KK_{*+2}$), and how to see that?

Similarly, a graded irreducible representation of $\mathbb{C}_{2k}$ gives a Morita equivalence between $\mathbb{C}$ and $\mathbb{C}_{2k}$. Does it induce the periodicity isomorphism, and if yes, which of the two gradings should one use? (relationship to the above question: $\varphi$ is easily seen to be the $KK$-inverse of the standard graded irreducible representation $W$ of $\mathbb{C}_2$ where the complex volume element implements the grading, since then $\varphi(1)$ projects onto the even part of $W$)?

choosesuch a Morita quivalence in $KK(C_2,0)$ and then exterior Kasparov multiplication with this element implements 2-periodicity. $\endgroup$