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

Abstract

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.