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

Abstract

Suppose that $\alpha$ is an action of the semigroup ${\mathbb{N}}^2$ on a $C^{\ast }$-algebra $A$ by endomorphisms. Let $A{\times }_{\alpha }^{\text{piso}}{\mathbb{N}}^2$ 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 $0\le L_1\le L_2\le A{\times }_{\alpha }^{\text{piso}}{\mathbb{N}}^2$ of essential ideals is obtained for which we identify the subquotients with familiar algebras.

arXiv commentThis is the revised version, 24 pages. arXiv admin note: text overlap with arXiv:1912.09682