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

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Let Γ⁺ be the positive cone of a totally ordered abelian discrete group Γ, and α an action of Γ⁺ by extendible endomorphisms of a C*-algebra A. We prove that the partial-isometric crossed product A×αpisoΓ⁺ is a full corner of a group crossed product B×βΓ, where B is a subalgebra of ℓ∞(Γ,A) generated by a collection of faithful copies of A, and the action β on B is induced by shift on ℓ∞(Γ,A). We then use this realization to show that A×αpisoΓ⁺ has an essential ideal J, which is a full corner in an ideal I×βΓ of B×βΓ.