Let r+ be the positive cone in a totally ordered abelian group F. We construct crossed products by actions of r1" as endomorphisms of Calgebras, and give criteria which ensure a given representation of the crossed product is faithful. We use this to prove that the C*-algebras generated by two semigroups V, W : P"-» B{H) of nonunitary isometries are canonically isomorphic, thus giving a new, self-contained proof of a theorem of Murphy, which includes earlier results of Coburn and Douglas.

Let r+ be the positive cone in a totally ordered abelian group F. We construct crossed products by actions of r1" as endomorphisms of Calgebras, and give criteria which ensure a given representation of the crossed product is faithful. We use this to prove that the C* -algebras generated by two semigroups V, W : P" -» B{H) of nonunitary isometries are canonically isomorphic, thus giving a new, self-contained proof of a theorem of Murphy, which includes earlier results of Coburn and Douglas.

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 be a totally ordered abelian group and I an order ideal in . We prove a theorem which relates the structure of the Toeplitz algebra T ( ) to the structure of the Toeplitz algebras T (I) and T ( /I). We then describe the primitive ideal space of the Toeplitz algebra T ( ) when the set ( ) of order ideals in is well-ordered, and use this together with our structure theorem to deduce information about the ideal structure of T ( ) when 0 → I → → /I → 0 is a non-trivial group extension.2000 Mathematics Subject Classification. 46L55.Introduction. Let be a totally ordered abelian group with positive cone + , and denote by {e x : x ∈ + } the usual basis for the Hilbert space 2 ( + ). For each x ∈ + , there is an isometry T x on 2 ( + ) such that T x e y = e x+y for all y ∈ + .The Toeplitz algebra of is the C * -subalgebra T ( ) of B( 2 ( + )) generated by the isometries {T x : x ∈ + }. These Toeplitz algebras include as special cases the algebras studied by Coburn [7] and Douglas [8], and generalisations to various classes of partially ordered groups have attracted a great deal of attention in recent years (see [12,13,10,11], for example).In [4], we considered the problem of describing the ideal structure of T ( ), and found that a crucial ingredient is the set ( ) of order ideals in , which is itself totally ordered under inclusion. We showed that the primitive ideals of T ( ) are parametrised by the disjoint union X( ) := { I : I ∈ ( )} of the duals of the discrete abelian

We investigate the ideal structure of the Toeplitz algebra T (Γ) of a totally ordered abelian group Γ. We show that the primitive ideals of T (Γ) are parametrised by the disjoint union X of the dualsÎ of the order ideals I of Γ, and identify the hull-kernel topology on X when the chain of order ideals in Γ is isomorphic to a subset of {−∞} ∪ Z ∪ {∞}.