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

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