Semigroup crossed products and the induced algebras of lattice-ordered groups

Let (G,G+) be a quasi-lattice-ordered group with positive cone G+. Laca and Raeburn have shown that the universal C∗-algebra C∗(G,G+) introduced by Nica is a crossed product BG+×αG+ by a semigroup of endomorphisms. The goal of this paper is to extend some results for totally ordered abelian groups to the case of discrete lattice-ordered abelian groups. In particular given a hereditary subsemigroup H+ of G+ we introduce a closed ideal IH+ of the C∗-algebra BG+. We construct an approximate identity for this ideal and show that IH+ is extendibly α-invariant. It follows that there is an isomorphism between C∗-crossed products and B(G/H)+×βG+. This leads to our main result that B(G/H)+×βG+ is realized as an induced C∗-algebra .