Let $A$ be a square matrix and let $\mathcal{O}_A$ be the corresponding Cuntz-Krieger algebra. Let furthermore $\gamma \colon \mathbb{T} \to \text{Aut}(\mathcal{O}_A)$ be the canonical guage-action and let $F_A^{\gamma}$ and $\mathbb{K}$ be the fixed-point algebra in $\mathcal{O}_A$ and the compact operators, respectively. I have read somewhere that one may regard the stabilisation of $\mathcal{O}_A$ as a crossed product with respect to the action:
$\mathcal{O}_A \otimes \mathbb{K} = F_A^{\gamma} \times_{\gamma} \mathbb{Z}$
Is this correct, and if yes, does anyone have a reference/proof of the result? I would simply like to point to some article stating the result so that I can take advantage of it myself.