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

Abstract: 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 this family to give new definitions of the reduced and essential crossed product C*-algebras that avoid, respectively, passing to the double commutant and local multiplier algebra of $A$. Our reduced crossed product is isomorphic to the one defined by Exel, Buss and Meyer, and when the inverse semigroup is quasi-countable our essential crossed product is isomorphic to the one defined by Kwaśniewski and Meyer.