THE PARTIAL-ISOMETRIC CROSSED PRODUCTS BY SEMIGROUPS OF ENDOMORPHISMS AS FULL CORNERS

Abstract

AbstractSuppose that ${\Gamma }^{+ } $ is the positive cone of a totally ordered abelian group $\Gamma $, and $(A, {\Gamma }^{+ } , \alpha )$ is a system consisting of a ${C}^{\ast } $-algebra $A$, an action $\alpha $ of ${\Gamma }^{+ } $ by extendible endomorphisms of $A$. We prove that the partial-isometric crossed product $A\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{piso} } \hspace{0.167em} {\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)\hspace{0.167em} {\mathop{\times }\nolimits }^{\mathrm{iso} } \hspace{0.167em} {\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\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{piso} } \hspace{0.167em} {\Gamma }^{+ } $ such that the quotient is the isometric crossed product $A\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{iso} } \hspace{0.167em} {\Gamma }^{+ } $.