Suppose $\Gamma^{ +}$ is the positive cone of a totally ordered abelian group $\Gamma$, and $(A,\Gamma^{ +},\alpha)$ is a system consisting of a $C^*$-algebra $A$, an action $\alpha$ of $\Gamma^{ +}$ by extendible endomorphisms of $A$. We prove that the partial-isometric crossed product $A\times_{ \alpha}^{ \piso}\Gamma^{ +}$ is a full corner in the subalgebra of $\L(\ell^{ 2}(\Gamma^{ +},A))$, and that if $\alpha$ is an action by automorphisms of $A$, then it is the isometric-crossed product $(B_{ \Gamma^{ +}}\otimes A)\times^{ \iso}\Gamma^{ +}$, 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 $A\times_{ \alpha}^{ \piso}\Gamma^{ +}$ such that the quotient is the isometric crossed product $A\times_{ \alpha}^{ \iso}\Gamma^{ +}$.
Categories: math.OA