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

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 \(A\times_{\alpha}^{\iso}\Gamma^{+}\).