Mamoon Ahmed. Source: Annals of Functional Analysis, Volume 10, Number 3, 370--380.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 $C^{*}$ -subalgebra $B_{G_{+}}$ of $\ell ^{\infty }(G_{+})$ spanned by the functions $\{1_{x}:x\in G_{+}\}$ . Then we showed that the crossed product $C^{*}$ -algebra $B_{(G/H)_{+}}\times _{\beta}G_{+}$ is realized as an induced $C^{*}$ -algebra $\operatorname{Ind}^{\widehat{G}}_{H^{\bot }}(B_{(G/H)_{+}}\times _{\tau }(G/H)_{+})$ . In this paper, we prove the existence of the following short exact sequence of $C^{*}$ -algebras: \begin{equation*}0\to I_{H_{+}}\times _{\alpha }G_{+}\to B_{G_{+}}\times _{\alpha }G_{+}\to \operatorname{Ind}^{\widehat{G}}_{H^{\bot }}(B_{(G/H)_{+}}\times _{\tau }(G/H)_{+})\to 0.\end{equation*} 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 $C^{*}$ -crossed products by hereditary subsemigroups involving an extension of previous results by Adji, Raeburn, and Rosjanuardi.