A new look at the crossed product of a C*-algebra by a semigroup of endomorphisms

Let G be a group and let P subset of G be a subsemigroup. In order to describe the crossed product of a C*-algebra A by an action of P by unital endomorphisms we find that we must extend the action to the whole group G. This extension fits into a broader notion of interaction groups which consists of an assignment of a positive operator V-g on A for each g in G, obeying a partial group law, and such that (V-g, V-g-1) is an interaction for every g, as defined in a previous paper by the author. We then develop a theory of crossed products by interaction groups and compare it to other endomorphism crossed product constructions.