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 + .

Let (A, α) be a system consisting of a C * -algebra A and an automorphism α of A. We describe the primitive ideal space of the partial-isometric crossed product A × piso α N of the system by using its realization as a full corner of a classical crossed product and applying some results of Williams and Echterhoff.2010 Mathematics Subject Classification. Primary 46L55.

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.

Let Γ + be the positive cone of a totally ordered abelian discrete group Γ, and α an action of Γ + by extendible endomorphisms of a C * -algebra A. We prove that the partial-isometric crossed product A × piso α Γ + is a full corner of a group crossed product B × β Γ, where B is a subalgebra of ℓ ∞ (Γ, A) generated by a collection of faithful copies of A, and the action β on B is induced by shift on ℓ ∞ (Γ, A). We then use this realization to show that A × piso α Γ + has an essential ideal J, which is a full corner in an ideal I × β Γ of B × β Γ. α Γ + is a full corner in a subalgebra of adjointable operators on ℓ 2 (Γ + ) ⊗ A ≃ ℓ 2 (Γ + , A), and when the action α is given by semigroup of automorphisms, A × piso α Γ + is a full corner in 2010 Mathematics Subject Classification. Primary 46L55. Key words and phrases. C * -algebra, endomorphism, semigroup, partial isometry, crossed product.

Let P be a left LCM semigroup, and α an action of P by endomorphisms of a C * -algebra A. We study a semigroup crossed product C * -algebra in which the action α is implemented by partial isometries. This crossed product gives a model for the Nica-Teoplitz algebras of product systems of Hilbert bimodules (associated with semigroup dynamical systems) studied first by Fowler, for which we provide a structure theorem as it behaves well under short exact sequences and tensor products.