THE COMPOSITION SERIES OF IDEALS OF THE PARTIAL-ISOMETRIC CROSSED PRODUCT BY SEMIGROUP OF ENDOMORPHISMS

Title & Authors
THE COMPOSITION SERIES OF IDEALS OF THE PARTIAL-ISOMETRIC CROSSED PRODUCT BY SEMIGROUP OF ENDOMORPHISMS

Abstract
Let $\small{{\Gamma}^+}$ be the positive cone in a totally ordered abelian group $\small{{\Gamma}}$, and $\small{{\alpha}}$ an action of $\small{{\Gamma}^+}$ by extendible endomorphisms of a $\small{C^*}$-algebra A. Suppose I is an extendible $\small{{\alpha}}$-invariant ideal of A. We prove that the partial-isometric crossed product $\small{\mathcal{I}:=I{\times}^{piso}_{\alpha}{\Gamma}^+}$ embeds naturally as an ideal of $\small{A{\times}^{piso}_{\alpha}{\Gamma}^+}$, such that the quotient is the partial-isometric crossed product of the quotient algebra. We claim that this ideal $\small{\mathcal{I}}$ together with the kernel of a natural homomorphism $\small{\phi:A{\times}^{piso}_{\alpha}{\Gamma}^+{\rightarrow}A{\times}^{iso}_{\alpha}{\Gamma}^+}$ gives a composition series of ideals of $\small{A{\times}^{piso}_{\alpha}{\Gamma}^+}$ studied by Lindiarni and Raeburn.
Keywords
$\small{C^*}$-algebra;endomorphism;semigroup;partial isometry;crossed product;primitive ideal;hull-kernel closure;
Language
English
Cited by
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