Crossed Products by Semigroups of Endomorphisms and the Toeplitz Algebras of Ordered Groups

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