The composition series of ideals of the partial-isometric crossed product by the semigroup $\mathbb{N}^{2}$

TL;DR

This paper studies the ideal structure of the partial-isometric crossed product of a $C^*$-algebra by the semigroup $ ^2$, revealing a composition series and identifying subquotients with known algebras.

Contribution

It provides a detailed composition series of the ideals in the partial-isometric crossed product by $ ^2$, connecting it to crossed products by $z^2$ and identifying subquotients.

Findings

Established a composition series of essential ideals

Identified subquotients with familiar $C^*$-algebras

Embedded the semigroup crossed product into a group crossed product

Abstract

Suppose that $\alpha$ is an action of the semigroup $\mathbb{N}^{2}$ on a $C^*$-algebra $A$ by endomorphisms. Let $A\times_{\alpha}^{\textrm{piso}} \mathbb{N}^{2}$ be the associated partial-isometric crossed product. By applying an earlier result which embeds this semigroup crossed product (as a full corner) in a crossed product by the group $\mathbb{Z}^{2}$, a composition series $0\leq L_{1}\leq L_{2}\leq A\times_{\alpha}^{\textrm{piso}} \mathbb{N}^{2}$ of essential ideals is obtained for which we identify the subquotients with familiar algebras.