Journal of the
Korean Mathematical Society JKMS

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 ${\mathcal I}:=I\times_{\alpha}^{\piso}\Gamma^{+}$ embeds naturally as an ideal of $A\times_{\alpha}^{\piso}\Gamma^{+}$, 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 $\phi: A\times_{\alpha}^{\piso}\Gamma^{+}\rightarrow A\times_{\alpha}^{\iso}\Gamma^{+}$ gives a composition series of ideals of $A\times_{\alpha}^{\piso}\Gamma^{+}$ studied by Lindiarni and Raeburn.

Keywords: $C^*$-algebra, endomorphism, semigroup, partial isometry, crossed product, primitive ideal, hull-kernel closure

MSC numbers: Primary 46L55