Suppose Γ^+ is the positive cone of a totally ordered abelian group Γ, and (A,Γ^+,α) is a system consisting of a C^*-algebra A, an action α of Γ^+ by extendible endomorphisms of A. We prove that the partial-isometric crossed product A×_α^Γ^+ is a full corner in the subalgebra of Ł(ℓ^2(Γ^+,A)), and that if α is an action by automorphisms of A, then it is the isometric-crossed product (B_Γ^+⊗ A)×^Γ^+, which is therefore a full corner in the usual crossed product of system by a group of

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... ms. We use these realizations to identify the ideal of A×_α^Γ^+ such that the quotient is the isometric crossed product A×_α^Γ^+.