Actions of inverse semigroups arising from partial actions of groups

In this work we present a definition of crossed product for actions of inverse semigroups on C ∗ -algebras, without resorting to covariant representations as done by Sieben in related work. We also show the existence of an isomorphism between the crossed product by a partial action of a group G and the crossed product by a related action of S ( G ) , an inverse semigroup associated to G introduced by the first named author.