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 B G + × α G + 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 I H + of the C ∗ -algebra B G + . We construct an approximate identity for this ideal and show that I H + is extendibly α-invariant. It follows that there is an isomorphism between C ∗ -crossed products ( B G + / I H + ) × α ˜ G + and B ( G / H ) + × β G + . This leads to our main result that B ( G / H ) + × β G + is realized as an induced C ∗ -algebra Ind H ⊥ G ˆ ( B ( G / H ) + × τ ( G / H ) + ) .