The composition series of ideals of the partial-isometric crossed product by semigroup of endomorphisms

Abstract

Let ${\Gamma }^{+}$ be the positive cone in a totally ordered abelian group $\Gamma$, and $\alpha$ an action of ${\Gamma }^{+}$ by extendible endomorphisms of a $C^{\ast }$-algebra $A$. Suppose $I$ is an extendible $\alpha$-invariant ideal of $A$. We prove that the partial-isometric crossed product $\mathcal{I}:=I{\times }_{\alpha }^{\text{piso}}{\Gamma }^{+}$ embeds naturally as an ideal of $A{\times }_{\alpha }^{\text{piso}}{\Gamma }^{+}$, such that the quotient is the partial-isometric crossed product of the quotient algebra. We claim that this ideal $\mathcal{I}$ together with the kernel of a natural homomorphism $\varphi :A{\times }_{\alpha }^{\text{piso}}{\Gamma }^{+}\to A{\times }_{\alpha }^{\text{iso}}{\Gamma }^{+}$ gives a composition series of ideals of $A{\times }_{\alpha }^{\text{piso}}{\Gamma }^{+}$ studied by Lindiarni and Raeburn.

The paper is going to appear in J. Korean Math. Soc