The Nica-Toeplitz algebras of dynamical systems over abelian lattice-ordered groups as full corners

Consider the pair $(G,P)$ consisting of an abelian lattice-ordered discrete group $G$ and its positive cone $P$. Let $\alpha$ be an action of $P$ by extendible endomorphisms of a $C^*$-algebra $A$. We show that the Nica-Toeplitz algebra $\mathcal{T}_{\textrm{cov}}(A\times_{\alpha} P)$ is a full corner of a group crossed product $\mathcal{B}\rtimes_{\beta}G$, where $\mathcal{B}$ is a subalgebra of $\ell^{\infty}(G,A)$ generated by a collection of faithful copies of $A$, and the action $\beta$ on $\mathcal{B}$ is given by the shift on $\ell^{\infty}(G,A)$. By using this realization, we show that the ideal $\mathcal{I}$ of $\mathcal{T}_{\textrm{cov}}(A\times_{\alpha} P)$ for which the quotient algebra $\mathcal{T}_{\textrm{cov}}(A\times_{\alpha} P)/\mathcal{I}$ is the isometric crossed product $A\times_{\alpha}^{\textrm{iso}} P$ is also a full corner in an ideal $\mathcal{J}\rtimes_{\beta}G$ of $\mathcal{B}\rtimes_{\beta}G$.