The primitive ideal space of the partial-isometric crossed product by automorphic actions of the semigroup ${\mathbb{N}}^2$

Let $(A,{\mathbb{N}}^2,\alpha )$ be a dynamical system consisting of a $C^{\ast }$-algebra $A$ and an action $\alpha$ of ${\mathbb{N}}^2$ on $A$ by automorphisms. Let $A{\times }_{\alpha }^{\text{piso}}{\mathbb{N}}^2$ be the partial-isometric crossed product of the system. We apply the fact that it is a full corner of a crossed product by the group ${\mathbb{Z}}^2$ in order to give a complete description of its primitive ideal space.