Rigidity of Generalized Furstenberg Boundaries and Applications to Intermediate Crossed Products
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918276471193600 |
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| author | Amrutam, Tattwamasi Liu, Chunlin |
| author_facet | Amrutam, Tattwamasi Liu, Chunlin |
| contents | We develop a relative boundary theory for actions of discrete groups on compact spaces and use it to derive rigidity results for reduced crossed products. For a discrete group $Γ$ acting on a compact space $X$ and a subgroup $H$, we construct a universal boundary over $X$ which is minimal as a $Γ$-system and strongly proximal with respect to $H$. When $H\le_cΓ$ is commensurated and the $H$-action on $X$ is minimal, we show that this universal boundary agrees, in a canonical $Γ$-equivariant way, with the generalized Furstenberg boundary of $(H,X)$, thereby unifying and extending earlier results on relative boundaries.
As an application, we introduce the notion of an $X$-plump subgroup given a $Γ$-space $X$, a generalized version of plumpness tailored to crossed products. Under natural dynamical hypotheses, this leads to new examples of irreducible $C^*$-inclusions. Under additional assumptions, we also show that every intermediate $C^*$-algebra is a crossed product. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03952 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rigidity of Generalized Furstenberg Boundaries and Applications to Intermediate Crossed Products Amrutam, Tattwamasi Liu, Chunlin Operator Algebras Dynamical Systems Group Theory We develop a relative boundary theory for actions of discrete groups on compact spaces and use it to derive rigidity results for reduced crossed products. For a discrete group $Γ$ acting on a compact space $X$ and a subgroup $H$, we construct a universal boundary over $X$ which is minimal as a $Γ$-system and strongly proximal with respect to $H$. When $H\le_cΓ$ is commensurated and the $H$-action on $X$ is minimal, we show that this universal boundary agrees, in a canonical $Γ$-equivariant way, with the generalized Furstenberg boundary of $(H,X)$, thereby unifying and extending earlier results on relative boundaries. As an application, we introduce the notion of an $X$-plump subgroup given a $Γ$-space $X$, a generalized version of plumpness tailored to crossed products. Under natural dynamical hypotheses, this leads to new examples of irreducible $C^*$-inclusions. Under additional assumptions, we also show that every intermediate $C^*$-algebra is a crossed product. |
| title | Rigidity of Generalized Furstenberg Boundaries and Applications to Intermediate Crossed Products |
| topic | Operator Algebras Dynamical Systems Group Theory |
| url | https://arxiv.org/abs/2601.03952 |