A dichotomy for inverse-semigroup crossed products via dynamical Cuntz semigroups

We characterise stable finiteness and pure infiniteness of the essential crossed product of a C*-algebra by an action of an inverse semigroup.

Under additional assumptions, we prove a stably finite / purely infinite dichotomy.

Our main technique is the development, using an induced action, of a ``dynamical Cuntz semigroup'' that is a subquotient of the usual Cuntz semigroup.

We prove that the essential crossed product is stably finite / purely infinite if and only if the dynamical Cuntz semigroup admits / does not admit a nontrivial state.

Indeed, a retract of our dynamical Cuntz semigroup suffices to prove the dichotomy.

Our results generalise those by Rainone on crossed products of groups acting by automorphisms of a C*-algebra, and we recover results by Kwaśniewski--Meyer--Prasad on C*-algebras of non-Hausdorff groupoids.