Covariant representations of actions of inverse semigroups: a new approach to the reduced and essential crossed-product C*-algebras

We consider an action of an inverse semigroup on a C*-algebra $A$ and use it to construct a groupoid of germs with unit space the spectrum of $A$. Motivated by the representation theory of C*-algebras of groupoids, we construct a concrete family of covariant representations for the action. We use th