The partial-isometric crossed products by semigroups of endomorphisms are Morita equivalent to crossed products by groups

Let $\Gamma^{+}$ be the positive cone of a totally ordered abelian discrete group $\Gamma$, and $\alpha$ an action of $\Gamma^{+}$ by extendible endomorphisms of a $C^*$-algebra $A$. We prove that the partial-isometric crossed product $A\times_{\alpha}^{\textrm{piso}}\Gamma^{+}$ is a full corner of a group crossed product $\mathcal{B}\times_{\beta}\Gamma$, where $\mathcal{B}$ is a subalgebra of $\ell^{\infty}(\Gamma,A)$ generated by a collection of faithful copies of $A$, and the action $\beta$ on $\mathcal{B}$ is induced by shift on $\ell^{\infty}(\Gamma,A)$. We then use this realization to show that $A\times_{\alpha}^{\textrm{piso}}\Gamma^{+}$ has an essential ideal $J$, which is a full corner in an ideal $\mathcal{I}\times_{\beta}\Gamma$ of $\mathcal{B}\times_{\beta}\Gamma$.