The Partial-Isometric Crossed Products by Semigroups of Endomorphisms as Full Corners

Suppose ${\mathrm{\Gamma }}^{+}$ is the positive cone of a totally ordered abelian group $\mathrm{\Gamma }$, and $(A,{\mathrm{\Gamma }}^{+},\alpha )$ is a system consisting of a $C^{\ast }$-algebra $A$, an action $\alpha$ of ${\mathrm{\Gamma }}^{+}$ by extendible endomorphisms of $A$. We prove that the partial-isometric crossed product $A{\times }_{\alpha }^{\text{piso}}{\mathrm{\Gamma }}^{+}$ is a full corner in the subalgebra of $\text{L}({\mathrm{\ell }}^2({\mathrm{\Gamma }}^{+},A))$, and that if $\alpha$ is an action by automorphisms of $A$, then it is the isometric-crossed product $(B_{{\mathrm{\Gamma }}^{+}}\otimes A){\times }^{\text{iso}}{\mathrm{\Gamma }}^{+}$, which is therefore a full corner in the usual crossed product of system by a group of automorphisms. We use these realizations to identify the ideal of $A{\times }_{\alpha }^{\text{piso}}{\mathrm{\Gamma }}^{+}$ such that the quotient is the isometric crossed product $A{\times }_{\alpha }^{\text{iso}}{\mathrm{\Gamma }}^{+}$.