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

Abstract

Let ${\Gamma }^{+}$ be the positive cone of a totally ordered abelian discrete group $\Gamma$, and $\alpha$ an action of ${\Gamma }^{+}$ by extendible endomorphisms of a $C^{\ast }$-algebra $A$. We prove that the partial-isometric crossed product $A{\times }_{\alpha }^{\text{piso}}{\Gamma }^{+}$ is a full corner of a group crossed product $\mathcal{B}{\times }_{\beta }\Gamma$, where $\mathcal{B}$ is a subalgebra of ${\ell }^{\infty }(\Gamma ,A)$ generated by a collection of faithful copies of $A$, and the action $\beta$ on $\mathcal{B}$ is induced by shift on ${\ell }^{\infty }(\Gamma ,A)$. We then use this realization to show that $A{\times }_{\alpha }^{\text{piso}}{\Gamma }^{+}$ has an essential ideal $J$, which is a full corner in an ideal $\mathcal{I}{\times }_{\beta }\Gamma$ of $\mathcal{B}{\times }_{\beta }\Gamma$.

References in corpus (1)

• Partial-isometric crossed products by semigroups of endomorphisms