A Dichotomy for Inverse-Semigroup Crossed Products via Dynamical Cuntz Semigroups

TL;DR

This paper characterizes when the essential crossed product of a C*-algebra by an inverse semigroup action is stably finite or purely infinite, using a new dynamical Cuntz semigroup concept, generalizing previous results.

Contribution

It introduces a dynamical Cuntz semigroup to analyze crossed products by inverse semigroup actions, establishing a dichotomy criterion based on states of this semigroup.

Findings

Essential crossed product is stably finite if the dynamical Cuntz semigroup admits a nontrivial state.

Essential crossed product is purely infinite if the dynamical Cuntz semigroup admits no nontrivial state.

Generalizes previous results by Rainone and Kwaśniewski--Meyer--Prasad.

Abstract

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 Kwa\'sniewski--Meyer--Prasad on C*-algebras of non-Hausdorff groupoids.