Crossed Products by Semigroups of Endomorphisms and Groups of Partial Automorphisms

We consider a class (A, S, alpha) of dynamical systems, where S is an Ore semigroup and alpha is an action such that each alphas is injective and extendible ( i.e. it extends to a non-unital endomorphism of the multiplier algebra), and has range an ideal of A. We show that there is a partial action on the fixed-point algebra under the canonical coaction of the enveloping group G of S constructed in [Proposition 6.1]{LR2}. It turns out that the full crossed product by this coaction is isomorphic to A rtimes alpha S. If the coaction is moreover normal, then the isomorphism can be extended to include the reduced crossed product. We look then at invariant ideals and finally, at examples of systems where our results apply.