The ideal structure of \(C^\ast\)-algebras related to lattice-ordered groups (Q2150315)

The paper by Mamoon Ahmed has the following 4 sections. 1 Introduction. In particular, given are a lattice-ordered abelian group \((G, G_+)\) with positive cone semigroup \(G_+\) and its \(C^*\)-algebra \(B_{G_+} \rtimes_{\alpha} G_+\) as the crossed product \(C^*\)-algebra of the semigroup dynamical system \((B_{G_+}, G_+,\alpha)\) of the \(C^*\)-algebra \(B_{G_+}\) generated by characteristic functions associated to elements of \(G_+\) by an action \(\alpha\) of \(G_+\), as given by Laca and Raeburn and as shown by Ahmed and Pryde, and by Ahmed. 2 Definitions and background material. In particular, given are the definitions for partially ordered groups \((G, G_+)\) and those for a semigroup dynamical system and its crossed product \(C^*\)-algebra. 3 The composition and primitive ideals. In particular, the composition is triply composed as the surjective homomorphism from \(B_{G_+}\rtimes_{\alpha} G_+\) to \(B_{(G/H)_+} \rtimes_{\beta}G_+\), the isomorphism implemented by the dual action \(\hat{\beta}\) of \(\beta\) by the quotient by \(H\) a subgroup of \(G\), and the surjective homomorphism from \(B_{(G/H)_+} \rtimes_{\beta} G_+\) to \(B_{(G/H)_+}\rtimes_{\gamma} (G/H)_+\) composed in this order. The primitive ideal structure in the case of totally ordered abelian groups has been considered by Adji and Raeburn. 4 Examples. In particular, given are the case for \((\mathbb{Z},\mathbb{N})\) of the group of integers and the semigroup of natural numbers as \((G, G_+)\) and that for \((\mathbb{Z}^2, \mathbb{N}^2)\).