The Partial-Isometric Crossed Products by Semigroups of Endomorphisms as Full Corners

Abstract

Suppose Γ + is the positive cone of a totally ordered abelian group Γ, and (A, Γ + , α) is a system consisting of a C * -algebra A, an action α of Γ + by extendible endomorphisms of A. We prove that the partial-isometric crossed product A × piso α Γ + is a full corner in the subalgebra of L(ℓ 2 (Γ + , A)), and that if α is an action by automorphisms of A, then it is the isometric-crossed product (B Γ + ⊗ A) × iso Γ + , which is therefore a full corner in the usual crossed product of system by a group of automorphisms. We use these realizations to identify the ideal of A × piso α Γ + such that the quotient is the isometric crossed product A × iso α Γ + .

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

“…where R γ is a projection in M K ℓ 2 (N) ⊗ (A × iso γ N) = L(ℓ 2 (N) ⊗ (A × iso γ N)), and J is the essential of A × piso γ N such that (A × piso γ N)/J ≃ A × iso γ N (see again [3]). Therefore, This completes the proof.…”

“…(3.50) Moreover, since the systems (A, N, δ) and (A, N, γ) are obviously given by automorphic actions, the algebras D and C are isomorphic to B Z ⊗ A, where B Z is the subalgebra of ℓ ∞ (Z, A) generated by the characteristic functions {1 n : n ∈ Z} (see [23,Proposition 5.1]). Therefore, by [23,Corollary 5.2] or [3,Corollary 5.3], the algebras A × iso δ N and A × iso γ N are full corners in the group crossed products (B Z ⊗ A) ⋊ lt ⊗δ −1 Z and (B Z ⊗ A) ⋊ lt ⊗γ −1 Z, respectively. Corollary 3.12.…”

“…Note that, for the trivial system (C, N 2 , id), the short exact sequence (3.50), is just the well-known exact sequence 0 −→ C Z 2 −→ T (Z 2 ) −→ C(T 2 ) −→ 0, (3.52) where C Z 2 is the commutator ideal of the Toeplitz algebra T (Z 2 ) ≃ T (Z) ⊗ T (Z) (see also [22,Remark 5.6]). Moreover, the essential ideals I δ and I γ of C × piso id N 2 ≃ T (Z 2 ) are both isomorphic to the algebra K(ℓ 2 (N)) ⊗ (C × piso id N) ≃ K(ℓ 2 (N)) ⊗ T (Z), where C × piso id N ≃ T (Z) is known by [3,Example 4.3] (see also [22,Remark 5.4]). Also, in this case, we have…”

“…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].…”

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.…”