} \mathbb{I}_p$ and consider the ring ${\ensuremath{\mathit{KK}}\xspace}_*(\mathbb{I},\mathbb{I})$ with multiplication given by the Kasparov product. The non-unital subring $$\Lambda=\bigoplus_{p,q\in\mathcal{P}} {\ensuremath{\mathit{KK}}\xspace}_*(\mathbb{I}_p,\mathbb{I}_q)$$ of ${\ensuremath{\mathit{KK}}\xspace}_*(\mathbb{I},\mathbb{I})$ is called the ring of *Böckstein operations*. It consists of matrices indexed by $\mathcal{P} \times \mathcal{P}$ with only finitely many non-zero entries $\lambda_{pq}\in {\ensuremath{\mathit{KK}}\xspace}_*(\mathbb{I}_p,\mathbb{I}_q)$. The Kasparov product $${\ensuremath{\mathit{KK}}\xspace}_*(\mathbb{I}_p,\mathbb{I}_q)\times {\ensuremath{\mathit{KK}}\xspace}_*(\mathbb{I}_q, \mathfrak{A}) \to {\ensuremath{\mathit{KK}}\xspace}_*(\mathbb{I}_p, \mathfrak{A})$$ induces a natural $\Lambda$-module structure on the ${\ensuremath{\mathbb{Z}}\xspace}/2\times \mathcal{P}$-graded group $$\underline{K}( \mathfrak{A} )=\bigoplus_{p\in \mathcal{P}} K_*( \mathfrak{A};{\ensuremath{\mathbb{Z}}\xspace}_{p}).$$

If $\mathfrak{A}$ is a separable $C^{*}$-algebra over a totally disconnected, metrizable, compact space $X$, then $\underline{K}( \mathfrak{A} )$ has a natural structure of a module over the ring $C(X,\Lambda)$ of locally constant functions from $X$ to $\Lambda$,m and $K_{*} ( \mathfrak{A} )$ has a natural structure of a ${\ensuremath{\mathbb{Z}}\xspace}_2$-graded module over the ring $C(X,{\ensuremath{\mathbb{Z}}\xspace})$ of locally constant functions from $X$ to ${\ensuremath{\mathbb{Z}}\xspace}$. This is easily seen by observing that $ \mathfrak{A} \cong \bigoplus_{k=1}^n \mathfrak{A} (U_k)$ naturally for any clopen partition $(U_k)_{k=1}^n$ of $X$. In the case when we have an evenly graded homomorphism of ${\ensuremath{\mathbb{Z}}\xspace}_2$-graded $C(X,{\ensuremath{\mathbb{Z}}\xspace})$-modules, we will often abuse notation by just saying that we have a $C(X,{\ensuremath{\mathbb{Z}}\xspace})$-homomorphism.

Structural properties of $C^{*}$-algebras over $\widetilde{{\ensuremath{\mathbb{N}}\xspace}}$
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Let $\{ \mathfrak{A}_{n} \}_{n=1}^{\infty}$ be a sequence of $C^{*}$-algebras. Set

$\displaystyle c_{0} ( \{ 