The Primitive Ideal Space of the Partial-isometric Crossed Product by Automorphic Actions of the Semigroup $\mathbb{N}^{2}$

Abstract

Let $(A,\mathbb{N}^{2},\alpha)$ be a dynamical system consisting of a $C^{*}$-algebra $A$ and an action $\alpha$ of $\mathbb{N}^{2}$ on $A$ by automorphisms. Let $A \times_{\alpha}^{\mathrm{piso}} \mathbb{N}^{2}$ 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.