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

TL;DR

This paper characterizes the primitive ideal space of the partial-isometric crossed product of a $C^*$-algebra by automorphic actions of $ ^2$, using its relation to a full corner of a group crossed product.

Contribution

It provides a complete description of the primitive ideal space for the partial-isometric crossed product by automorphisms of $ ^2$, linking it to a group crossed product.

Findings

Primitive ideal space characterized explicitly.

Connection established between partial-isometric and group crossed products.

Complete description of the ideal structure achieved.

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 $A\times_{\alpha}^{\textrm{piso}}\mathbb{N}^{2}$ 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.