A Dichotomy for Inverse-Semigroup Crossed Products via Dynamical Cuntz Semigroups
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909987527196672 |
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| author | Armstrong, Becky Clark, Lisa Orloff Huef, Astrid An Martínez, Diego Tolich, Ilija |
| author_facet | Armstrong, Becky Clark, Lisa Orloff Huef, Astrid An Martínez, Diego Tolich, Ilija |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07100 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Dichotomy for Inverse-Semigroup Crossed Products via Dynamical Cuntz Semigroups Armstrong, Becky Clark, Lisa Orloff Huef, Astrid An Martínez, Diego Tolich, Ilija Operator Algebras 46L05 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. |
| title | A Dichotomy for Inverse-Semigroup Crossed Products via Dynamical Cuntz Semigroups |
| topic | Operator Algebras 46L05 |
| url | https://arxiv.org/abs/2601.07100 |