Hamiltonian diffeomorphisms of toric manifolds

We prove that ${\pi }_1(\text{Ham}(M))$ contains an infinite cyclic subgroup, where $\text{Ham}(M)$ is the Hamiltonian group of the one point blow up of $\mathbb{C}{P}^3$. We give a sufficient condition for the group ${\pi }_1(\text{Ham}(M))$ to contain an infinite cyclic subgroup, when $M$ is a general toric manifold.