Commutator ideals in C∗-crossed products by hereditary subsemigroups

Abstract

Let $(G,G_{+})$ be a lattice-ordered abelian group with positive cone $G_{+}$, and let $H_{+}$ be a hereditary subsemigroup of $G_{+}$. In previous work, the author and Pryde introduced a closed ideal $I_{H_{+}}$ of the $C^{\ast }$-subalgebra $B_{G_{+}}$ of ${\ell }^{\infty }\left(G_{+}\right)$ spanned by the functions $\{1_x:x\in G_{+}\}$. Then we showed that the crossed product $C^{\ast }$-algebra $B_{(G/H{)}_{+}}{\times }_{\beta }{G}_{+}$ is realized as an induced $C^{\ast }$-algebra ${Ind}_{H^{\perp }}^{\hat{G}}\left(B_{(G/H{)}_{+}}{\times }_{\tau }\right(G/H{)}_{+})$. In this paper, we prove the existence of the following short exact sequence of $C^{\ast }$-algebras: $$0\to I_{H_{+}}{\times }_{\alpha }{G}_{+}\to B_{G_{+}}{\times }_{\alpha }{G}_{+}\to {Ind}_{H^{\perp }}^{\hat{G}}\left(B_{(G/H{)}_{+}}{\times }_{\tau }\right(G/H{)}_{+})\to 0.$$ This relates $B_{G_{+}}{\times }_{\alpha }{G}_{+}$ to the structure of $I_{H_{+}}{\times }_{\alpha }{G}_{+}$ and $B_{(G/H{)}_{+}}{\times }_{\beta }{G}_{+}$. We then show that there is an isomorphism $\iota$ of $B_{H_{+}}{\times }_{\alpha }{H}_{+}$ into $B_{G_{+}}{\times }_{\alpha }{G}_{+}$. This leads to nontrivial results on commutator ideals in $C^{\ast }$-crossed products by hereditary subsemigroups involving an extension of previous results by Adji, Raeburn, and Rosjanuardi.