Haagerup property of semigroup crossed products

In this paper, we study the Haagerup property of semigroup crossed products for lattice ordered groups. Let G be a lattice ordered group. We show that the reduced semigroup C*-algebra C*(r) (G+) has the Haagerup property if and only if G has the Haagerup property. Let G act on a unital C*-algebra A through an action a, we show that if the reduced semigroup crossed product A xl(a,r) G+ has the Haagerup property, then both A and G have the Haagerup property. On the other hand, it is shown that if G is s-amenable and A has the Haagerup property, then A xl(a,r) G+ has the Haagerup property.