A Dichotomy for Inverse-Semigroup Crossed Products via Dynamical Cuntz Semigroups
Becky Armstrong, Lisa Orloff Clark, Astrid An Huef, Diego Martínez, Ilija Tolich
TL;DR
This work develops a dynamical Cuntz semigroup $Cu(A)_\alpha$ for actions $\alpha$ of inverse semigroups on C*-algebras and uses it to establish a stable finiteness / pure infiniteness dichotomy for the essential crossed product $A \rtimes_\text{ess} S$. The core technique is to compare Cu$(A)_\alpha$ with invariant functionals and traces, and to leverage retracts of Cu$(A)$ to obtain computable criteria; under mild hypotheses (e.g., $A$ exact and $A \rtimes_\text{red} S = A \rtimes_\text{ess} S$ with plain paradoxes), the crossed product is either stably finite or purely infinite, with equivalences mediated by the existence of nontrivial invariant functionals on Cu$(A)_\alpha$. The paper generalizes Rainone’s results for group actions and recovers Kwaśniewski–Meyer–Prasad’s groupoid dichotomy in the non-Hausdorff setting, and further applies the framework to groupoid C*-algebras via a retract-based approach, yielding explicit stably finite / purely infinite criteria in terms of lower-semicontinuous functionals on the retract Cu-semigroups. The results enhance the understanding of how dynamical structures control finiteness properties in crossed products, with concrete implications for C*-algebras of groupoids and their noncommutative dynamics.
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 Kwaśniewski--Meyer--Prasad on C*-algebras of non-Hausdorff groupoids.