The Composition Series of Ideals of the Partial-Isometric Crossed Product by Semigroup of Endomorphisms

Abstract

Abstract. Let Γ + be the positive cone in a totally ordered abelian group Γ, and α an action of Γ + by extendible endomorphisms of a C * -algebra A. Suppose I is an extendible α-invariant ideal of A. We prove that the partial-isometric crossed product I := I × piso α Γ + embeds naturally as an ideal of A × piso α Γ + , such that the quotient is the partial-isometric crossed product of the quotient algebra. We claim that this ideal I together with the kernel of a natural homomorphism φ : A × piso α Γ + → A × iso α Γ + gives a composition series of ideals of A × piso α Γ + studied by Lindiarni and Raeburn.

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

“…Note that, when P is the positive cone of a totally ordered abelian group G, the Nica covariance condition is satisfied automatically, and therefore A × piso α P is just the Toeplitz algebra T (X) (see [16]). Readers are referred to [1,3,4,13,23,22] to see further studies on the crossed product A × piso α P . Also, in [21], following the idea of [16], the notion of the partial-isometric crossed product of a system is extended to more general semigroups based on the works in [5,9,10].…”

Partial-isometric crossed products of dynamical systems by left LCM semigroups

“…When the group G is totally ordered and abelian (with the positive cone G + = P ), the Nica covariance condition holds automatically, and the Toeplitz algebra T (X) is the partial-isometric crossed product A × piso α P of the system (A, P, α) introduced and studied by the authors of [16]. In other word, the semigroup crossed product A × piso α P actually gives a model for the Teoplitz algebras T (X) of product systems X of Hilbert bimodules associated with the systems (A, P, α), where P is the positive cone of a totally ordered abelian group G. Further studies on the structure of the crossed product A × piso α P have been done progressively in [2], [3], [4], [13], and [22] since then.…”

The Nica-Toeplitz algebras of dynamical systems over abelian lattice-ordered groups as full corners

“…We recall that when the group G is totally ordered and abelian, the algebra T cov (A × α P ) is the partial-isometric crossed product A × piso α P of the system (A, P, α) introduced and studied in [13]. Further studies on the structure of the algebra A × piso α P were carried out in [1], [3], [4], [10], and [18]. In particular, it was shown in [18] that A × piso α P is a full corner in classical crossed product by group.…”