Crossed products by semigroups of endomorphisms and the Toeplitz algebras of ordered groups

Let Γ + {Γ ^ + } be the positive cone in a totally ordered abelian group Γ Γ . We construct crossed products by actions of Γ + {Γ ^ + } as endomorphisms of C ∗ {C^ } -algebras, and give criteria which ensure a given representation of the crossed product is faithful. We use this to prove that the C ∗ {C^ } -algebras generated by two semigroups V , W : Γ + → B ( H ) W:{Γ ^ + } → 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.